Large area power transmitter for wireless power transfer

ABSTRACT

A method for wireless power transfer. The method includes adapting a variable form factor transmitter into at least a plurality of cross-coupled segments disposed about a pre-determined wireless power transfer area, wherein the pre-determined wireless power transfer area comprises a dimension exceeding a wavelength corresponding to a characteristic frequency of the variable form factor transmitter, transmitting, from a radio frequency (RF) power source and based at least in part on the characteristic frequency, RF power across the pre-determined wireless power transfer area via a near electromagnetic field of the variable form factor transmitter, and reducing, based on opposing directions of magnetic fields induced by adjacent cross-coupled segments of the plurality of cross-coupled segments, a radiation loss of the wireless power transfer due to a far electromagnetic field of the variable form factor transmitter.

BACKGROUND

Wireless power transfer is the transmission of electrical energy from apower source to an electrical load without the use of man-madeconductors to connect the power source to the electrical load. Awireless power transfer system consists of a transmitter and one or morereceiver devices. The transmitter is connected to a source of power andconverts the power to a time-varying electromagnetic field. The one ormore receiver devices receive the power via the electromagnetic fieldand convert the received power back to an electric current to beutilized by the electrical load.

SUMMARY

In general, in one aspect, the invention relates to a method forwireless power transfer. The method includes adapting a variable formfactor transmitter into at least a plurality of cross-coupled segmentsdisposed about a pre-determined wireless power transfer area, whereinthe pre-determined wireless power transfer area comprises a dimensionexceeding a wavelength corresponding to a characteristic frequency ofthe variable form factor transmitter, transmitting, from a radiofrequency (RF) power source and based at least in part on thecharacteristic frequency, RF power across the pre-determined wirelesspower transfer area via a near electromagnetic field of the variableform factor transmitter, and reducing, based on opposing directions ofmagnetic fields induced by adjacent cross-coupled segments of theplurality of cross-coupled segments, a radiation loss of the wirelesspower transfer due to a far electromagnetic field of the variable formfactor transmitter.

In general, in one aspect, the invention relates to a variable formfactor transmitter for wireless power transfer. The variable form factortransmitter includes a plurality of cross-coupled segments disposedabout a pre-determined wireless power transfer area, wherein thepre-determined wireless power transfer area comprises a dimensionexceeding a wavelength corresponding to a characteristic frequency ofthe variable form factor transmitter, wherein the plurality ofcross-coupled segments are configured to transmit, from a radiofrequency (RF) power source and based at least in part on thecharacteristic frequency, RF power across the pre-determined wirelesspower transfer area via a near electromagnetic field of the variableform factor transmitter, and reduce, based on opposing directions ofmagnetic fields induced by adjacent cross-coupled segments of theplurality of cross-coupled segments, a radiation loss of the wirelesspower transfer due to a far electromagnetic field of the variable formfactor transmitter.

In general, in one aspect, the invention relates to a system forwireless power transfer. The system includes a plurality ofcross-coupled segments disposed about a pre-determined wireless powertransfer area, wherein the pre-determined wireless power transfer areacomprises a dimension exceeding a wavelength corresponding to acharacteristic frequency of the plurality of cross-coupled segments, anda radio frequency (RF) power source coupled to the plurality ofcross-coupled segments, wherein the plurality of cross-coupled segmentsare configured to transmit, from the RF power source and based at leastin part on the characteristic frequency, RF power across thepre-determined wireless power transfer area via a near electromagneticfield of the plurality of cross-coupled segments, and reduce, based onopposing directions of magnetic fields induced by adjacent cross-coupledsegments of the plurality of cross-coupled segments, a radiation loss ofthe wireless power transfer due to a far electromagnetic field of theplurality of cross-coupled segments.

Other aspects of the invention will be apparent from the followingtransaction description and the appended claims.

BRIEF DESCRIPTION OF DRAWINGS

FIGS. 1A, 1B, and 1C show schematic diagrams of an example system havinga variable form factor transmitter in accordance with one or moreembodiments of the invention.

FIGS. 2A, 2B, 2C, 2D, 2E, 2F, 2G, 2H, 2J, 2K, 2L, 2M, 2N, and 2P showvarious diagrams for illustrating an example variable form factortransmitter in accordance with one or more embodiments of the invention.

FIGS. 3A, 3B, 3C, 3D, and 3E show example characteristics of an examplevariable form factor transmitter in accordance with one or moreembodiments of the invention.

FIGS. 4A and 4B show schematic diagrams of example radio frequency (RF)power sources in accordance with one or more embodiments of theinvention.

FIGS. 5A, 5B, 5C, 5D, and 5E show schematic and layout diagrams ofexample receiver devices in accordance with one or more embodiments ofthe invention.

FIGS. 6A, 6B, 6C, 6D, 6E, 6F, 6G, 7A, 7B, 7C, 7D, 7E, and 7F showschematic and layout diagrams of example power transmitters for wirelesspower transfer in accordance with one or more embodiments of theinvention.

FIG. 8 shows a method flowchart in accordance with one or moreembodiments of the invention.

DETAILED DESCRIPTION

Specific embodiments of the invention will now be described in detailwith reference to the accompanying figures. Like elements in the variousfigures are denoted by like reference numerals for consistency.

In the following detailed description of embodiments of the invention,numerous specific details are set forth in order to provide a morethorough understanding of the invention. However, it will be apparent toone of ordinary skill in the art that the invention may be practicedwithout these specific details. In other instances, well-known featureshave not been described in detail to avoid unnecessarily complicatingthe description.

In the following description, any component described with regard to afigure, in various embodiments of the invention, may be equivalent toone or more like-named components described with regard to any otherfigure. For brevity, at least a portion of these components areimplicitly identified based on various legends. Further, descriptions ofthese components will not be repeated with regard to each figure. Thus,each and every embodiment of the components of each figure isincorporated by reference and assumed to be optionally present withinevery other figure having one or more like-named components.Additionally, in accordance with various embodiments of the invention,any description of the components of a figure is to be interpreted as anoptional embodiment which may be implemented in addition to, inconjunction with, or in place of the embodiments described with regardto a corresponding like-named component in any other figure. In thefigures, black solid collinear dots indicate that additional componentssimilar to the components before and/or after the solid collinear dotsmay optionally exist.

Throughout the application, ordinal numbers (e.g., first, second, third,etc.) may be used as an adjective for an element (i.e., any noun in theapplication). The use of ordinal numbers is not to imply or create anyparticular ordering of the elements nor to limit any element to beingonly a single element unless expressly disclosed, such as by the use ofthe terms “before”, “after”, “single”, and other such terminology.Rather, the use of ordinal numbers is to distinguish between theelements. By way of an example, a first element is distinct from asecond element, and the first element may encompass more than oneelement and succeed (or precede) the second element in an ordering ofelements.

In general, embodiments of the invention provide a method, a transmitterdevice, and a system for wireless power transfer. In one or moreembodiments of the invention, the method, transmitter device, and systemis based on a power transmitter that includes a number of capacitors andinductive segments disposed along a path that defines a wireless powertransfer area. The capacitors are connected in series via at least theinductive segments into a string of distributed capacitors. In one ormore embodiments of the invention, the string of distributed capacitorsis integrated onto a laminated material sheet (e.g., including at leastdielectric material) encompassing at least the path that defines thepower transfer area. In one or more embodiments, one or more capacitorsand one or more inductive segments are constructed using conductivestrips attached to two opposing surfaces of the material sheet. Forexample, a capacitor may include two conductive strips attached to thetwo opposing surfaces. Further, an inductive segment may include anotherconductive strip attached to one of the two opposing surfaces.Accordingly, RF power is transmitted, from a radio frequency (RF) powersource and based at least on a characteristic frequency of the string ofdistributed capacitors, across the wireless power transfer area via anear electromagnetic field of the of the string of distributedcapacitors. In one or more embodiments of the invention, thecharacteristic frequency is within the industrial, scientific andmedical (ISM) radio band defined by the International TelecommunicationUnion (ITU) Radio Regulations. For example, the characteristic frequencymay be within the type A frequency range (i.e., 6.765 MHz-6.795 MHz)defined in the ITU Radio Regulations Article 5, footnote 5.138.

In one or more embodiments of the invention, the power transmitter has afixed form factor specific to a pre-determined wireless power transferarea. In one or more embodiments of the invention, the power transmitteris adaptable into different form factors (referred to as adapted formfactors) to fit different wireless power transfer areas. In suchembodiments, the power transmitter is a variable form factor transmitterwith a characteristic frequency that is maintained to be substantiallyindependent of the adapted form factors. For example, the characteristicfrequency may be maintained, as the adapted form factor varies, withinthe ISM radio band.

FIG. 1A shows a schematic diagram of an example system (100) inaccordance with one or more embodiments of the invention. In one or moreembodiments, one or more of the modules and elements shown in FIG. 1Amay be omitted, repeated, and/or substituted. Accordingly, embodimentsof the invention should not be considered limited to the specificarrangements of modules shown in FIG. 1A.

As shown in FIG. 1A, the system (100) includes a variable form factortransmitter (102) receiving power from an RF power source (108) forwireless power transfer across a wireless power transfer area (101)having one or more receiver devices (e.g., denoted as circular iconslabeled as A, B, C, D, E, and F) disposed therein. Each of thesecomponents is described in detail below.

In one or more embodiments of the invention, the wireless power transferarea (101) is any three dimensional (3D) physical space where the one ormore receiver devices are receiving power from the variable form factortransmitter (102). For example, the wireless power transfer area (101)may include a 3D space within a building or a vehicle, such as a room, ahallway, a passenger cabin of a car, bus, train, airplane, or spaceship, or any portion of the building or vehicle. In another example, thewireless power transfer area (101) may include a 3D space that is notenclosed, such as a play ground, a roadway, an amusement park, or anytype of field on the ground, above the ground, or away from the earth inthe space (e.g., an atmospheric layer or interstellar space). In yetanother example, the wireless power transfer area (101) may include anunderground or under-water space, such as a cave, an under water regionnear an ocean platform or sea bed, etc. In still another example, thewireless power transfer area (101) may include a combination of theexamples above.

In one or more embodiments of the invention, the variable form factortransmitter (102) is disposed entirely within the wireless powertransfer area (101), overlaps the wireless power transfer area (101), orin the vicinity of the wireless power transfer area (101). In one ormore embodiments, at least a portion of the variable form factortransmitter (102) may be inserted in a protective sleeve, embedded in amaterial sheet, free-standing in the wireless power transfer area (101),or attached to the wireless power transfer area (101). In one or moreembodiments, at least a portion of the variable form factor transmitter(102) may be stationery or moving with respect to the wireless powertransfer area (101) and/or the one or more receiver devices (e.g.,denoted as circular icons labeled as A, B, C, D, E, and F) disposedtherein. In one or more embodiments of the invention, the form factor ofthe variable form factor transmitter (102) is adapted according to ageometric constraint imposed by the wireless power transfer area (101).For example, the variable form factor transmitter (102) may be made ofpliable material such that the form factor of the variable form factortransmitter (102) is changed by the user to fit the physical shape ofthe room, hallway, passenger cabin, play ground, roadway, amusementpark, field, cave, under-water region, etc. of the wireless powertransfer area (101). In this context, the form factor of the variableform factor transmitter (102) is based on the wireless power transferarea (101). For example, the form factor of the variable form factortransmitter (102) may include a 3D portion, such as a curved surface, ahelical curve, etc.

In one or more embodiments of the invention, the receiver devices (A)through (F) may be of the same type or of different types that are usedby one or more users, such as individual persons. In one or moreembodiments, one or more of the receiver devices (A) through (F) aredisposed at user specified locations throughout the wireless powertransfer area (101) and are stationary during the wireless powertransfer. In one or more embodiments, one or more of the receiverdevices (A) through (F) have dimensions that are smaller than thedimensions of the wireless power transfer area (101). In one or moreembodiments, one or more of the receiver devices (A) through (F) havedimensions that are comparable to or greater than the dimensions of thewireless power transfer area (101). For example, the receiver device (A)may be a lighting device placed on the ceiling of a room or hallway bythe user. In one or more embodiments, one or more of the receiverdevices (A) through (F) are carried by respective users who move aroundthroughout the wireless power transfer area (101) from time to timeduring the wireless power transfer. Based on the nature of the nearelectromagnetic field of the variable form factor transmitter (102), thepower of the near electromagnetic field that is not received by any ofthe receiver device is returned to the variable form factor transmitter(102) and the RF power source (108). This is in contrast to a farelectromagnetic field via which power is radiated resulting in energyloss that is not productive for the wireless power transfer. Examples ofthe receiver device (A), receiver device (B), receiver device (C),receiver device (D), receiver device (E), and receiver device (F) aredescribed in reference to FIGS. 5A, 5B, 5C, 5D, and 5E below.

In one or more embodiments of the invention, the variable form factortransmitter (102) includes a string of distributed capacitors. Inparticular, the string of distributed capacitors includes multiplecapacitor-wire segments that are connected in series to conductradio-frequency (RF) electrical current (105) generated by the powersource (108). The RF electrical current (105) induces magnetic fields(e.g., magnetic field (106)) that are present throughout the wirelesspower transfer area (101). In one or more embodiments, the string ofdistributed capacitors is disposed along a path such that the magneticfields throughout the wireless power transfer area (101) exceeds athreshold that is based on a power requirement of the receiver devices.In this context, the path is based on the wireless power transfer area(101). In one or more embodiments, the RF electrical current (105)enters/exits the wire at a terminal A (204 a) and a terminal B (204 b).In one or more embodiments, additional intervening components (notshown) may also be inserted in the series of capacitor-wire segments orinserted between the series of capacitor-wire segments and one or moreterminals (e.g., terminal A (204 a), terminal B (204 b)) withoutimpeding the operation of the variable form factor transmitter (102).

In one or more embodiments, each capacitor-wire segment includes acapacitor (e.g., capacitor (103)) connected to a wire segment (e.g.,wire segment (104)). In one or more embodiments, each capacitor (e.g.,capacitor (103)) in the variable form factor transmitter (102) has thesame nominal capacitance value, as any other capacitor therein, that isdetermined prior to disposing the variable form factor transmitter (102)in the wireless power transfer area (101). For example, the capacitors(e.g., capacitor (103)) in the variable form factor transmitter (102)may be installed in a factory before a user uses the variable formfactor transmitter (102) to provide power wirelessly within the wirelesspower transfer area (101). The capacitors (e.g., capacitor (103)) may beof a suitable type, such as ceramic capacitors, film and papercapacitors, electrolyte capacitors, polymer capacitors, silver micacapacitors, etc. In one or more embodiments, one or more of thecapacitors may include two aluminum or other metallic sheets, foils, orfilms separated by an aluminum or other metallic oxide layer. As istypical in a factory manufacturing process, the capacitance values ofall capacitors (e.g., capacitor (103)) in the variable form factortransmitter (102) may vary within a range (referred to as a capacitancerange), e.g., due to a manufacturing tolerance.

In one or more embodiments, each capacitor-wire segment includes a wiresegment having a pre-determined segment length and a pre-determinedinductance per unit length. For example, the wire segments (e.g., wiresegment (104)) in the variable form factor transmitter (102) may beinstalled in a factory before a user uses the variable form factortransmitter (102) to provide power wirelessly within the wireless powertransfer area (101). The wire segments (e.g., wire segment (104)) may beof a suitable type, such as insulated or un-insulated wires, sheets,foil, or films made of copper, aluminum, or other suitable metal and/oralloy material. In one or more embodiments, one or more of the wiresegments (e.g., wire segment (104)) are flexible or pliable such thatthe user may bend, stretch, or otherwise change the shape of the one ormore wire segments. As is typical in a factory manufacturing process,the length and inductance values of each and all wire segments (e.g.,wire segment (104)) in the variable form factor transmitter (102) mayvary within a range (referred to as a length range and an inductancerange), e.g., due to a manufacturing tolerance.

In one or more embodiments of the invention, by confining the electricalfields, the capacitors (e.g., capacitor (103)) in the variable formfactor transmitter (102) reduce stray electric fields and the resultantinduced voltage of the wire segments (e.g., wire segment (104)).Accordingly, the capacitors (e.g., capacitor (103)) in the variable formfactor transmitter (102) reduce the fraction of energy stored in thestray capacitance of the wire segments (e.g., wire segment (104)) overthe total energy in the system (100). The reduction of both inducedvoltage and stored energy associated with the stray capacitance reducesloss due to environmental interactions and improves safety for the user.

In one or more embodiments of the invention, the variable form factortransmitter (102) is associated with a characteristic frequency that isbased at least on the pre-determined capacitance, the pre-determinedsegment length, and the pre-determined inductance per unit length. Thecharacteristic frequency of the variable form factor transmitter (102)is described in reference to FIGS. 2A, 2B, 2D, 2E, 3A, 3B, 3C, 3D, and3E below. Throughout this document, the terms “characteristic frequency”and “resonant frequency” may be used interchangeably depending oncontext.

In one or more embodiments, instead of the direct connection to thepower source (108), the variable form factor transmitter (102) receivespower from the power source (108) using inductive coupling via a drivingloop (109 a). FIG. 1B shows a schematic diagram of the example system(100) in the inductive coupling power configuration. Details ofreceiving power via the driving loop (109 a) are described in referenceto FIG. 1C below.

FIG. 1C shows a schematic diagram of supplying power via the drivingloop (109 a) depicted in FIG. 1B above. In one or more embodiments, oneor more of the modules and elements shown in FIG. 1C may be omitted,repeated, and/or substituted. Accordingly, embodiments of the inventionshould not be considered limited to the specific arrangements of modulesshown in FIG. 1C.

As shown in FIG. 1C, the driving loop (109 a) includes one or more loopsof conducting wire (e.g., having an inductance L₁) that are coupled tothe power source (108) via a balun (108 a). The balun (108 a) includes atuning capacitor A (109 d) (e.g., having a variable capacitance C₁), atuning capacitor B (109 e) (e.g., having a variable capacitance C₂), anda coaxial cable (109 c) (e.g., coiled around a ferrite core (109 b) andhaving an inductance L₂). Specifically, the driving loop (109 a) isplaced at a distance (110) from the variable form factor transmitter(102) such that the power source (108) supplies power to the variableform factor transmitter (102) via electromagnetic coupling across thedistance (110). In one or more embodiments, the tuning capacitor B (109e) is tuned to resonate with inductance L₂ of the ferrite core (109 b)to form a parallel resonant LC circuit, which imposes a high impedancebetween the two opposite ends of the coaxial cable (109 c). Further, thetuning capacitor A (109 d) is used to tune the resonant frequency of thedriving loop (109 a) to match the frequency of the RF power source(108). The distance (110) between the driving loop (109 a) and thevariable form factor transmitter (102) may be adjusted in order to matchthe apparent input impedance of variable form factor transmitter (102)to the impedance of the coaxial cable (109 c), and the output impedanceof the RF power source (108).

FIG. 2A shows a schematic diagram of a parallel-wire transmission line(201) in accordance with one or more embodiments of the invention. Inone or more embodiments, one or more of the modules and elements shownin FIG. 2A may be omitted, repeated, and/or substituted. Accordingly,embodiments of the invention should not be considered limited to thespecific arrangements of modules shown in FIG. 2A.

As shown in FIG. 2A, the sinusoid-shaped icons (201 a) and (201 b)represent electromagnetic waves propagating along the parallel-wiretransmission line (201). The parallel-wire transmission line (201) iscomposed of two parallel wires (201 d) each having wire segments joinedby capacitors, where s denotes the length of each wire segment, Cdenotes the capacitance of each capacitor, and q denotes the electriccharge displacement along the parallel-wire transmission line (201). Inthe context that the two parallel wires (201 d) conduct RF current(e.g., electrical current (105) depicted in FIG. 1A), each wire of twoparallel wires (201 d) is also referred to as a conductor wirethroughout this document. The distance between the sinusoid-shaped icons(201 a) and (201 b) corresponds to the length of the parallel-wiretransmission line (201) while the spacing between the two parallelstring of capacitors corresponds to the width of the parallel-wiretransmission line (201). While the length of the parallel-wiretransmission line (201) may be comparable to a length of other dimensionof the wireless power transfer area (101), the width of theparallel-wire transmission line (201) may range from less than onecentimeter to a width or other dimension of the wireless power transferarea (101). In one or more embodiments, the parallel-wire transmissionline (201) corresponds to a portion of the variable form factortransmitter (102) depicted in FIG. 1A above. In other words, twosections of the string of distributed capacitor depicted in FIG. 1A maybe disposed parallel to each other. Generally, the electric charge, q,displaced along the parallel-wire transmission line (201) is a functionof a position along the parallel-wire transmission line (201) and time.The corresponding charge density (i.e., electric charge per unitlength), ρ_(λ), and electrical current, I, are given by Eq. (1) belowfor the parallel-wire transmission line (201). In Eq. (1), x and tdenote the position along the parallel-wire transmission line (201) andtime, respectively.

$\begin{matrix}{{\rho_{\lambda} = {q^{\prime} = \frac{\partial q}{\partial x}}},{I = {\overset{.}{q} = \frac{\partial q}{\partial t}}}} & {{Eq}.\mspace{20mu} (1)}\end{matrix}$

TABLE 1 shows additional definitions of variables used in the equationsthroughout this document.

TABLE 1   c = capacitance per milt length l = inductance per unit lengthC = capacitance of each joining capacitor s = length of each segment q =charge displacement ρ_(λ) = charge density λ = wavelength in free spaceI = current U_(j) = energy stored in the two joining capacitors u_(E) =electrical energy stored per unit length u_(B) = magnetic energy storedper unit length v = asymptotic velocity ω₀ = cutoff frequency v_(p) =phase velocity v_(g) = group veloctiy

The electrical energy, U_(j), stored in a pair of adjoining capacitors(e.g., capacitor pair (201 c)) in the parallel-wire transmission line(201) is given by Eq. (2) below.

$\begin{matrix}{U_{j} = {{{2 \cdot \frac{1}{2}}\frac{q^{2}}{C}} = \frac{q^{2}}{C}}} & {{Eq}.\mspace{14mu} (2)}\end{matrix}$

In the scenario where s is substantially less than the spatial variationof q, the stored energy, U_(j), divided by the segment length, s, may beconsidered as a density of energy stored in the capacitors, C, along theparallel-wire transmission line (201). Let c denote the straycapacitance per unit length between the two parallel wires of theparallel-wire transmission line (201). The total electrical energy,u_(E), stored per unit length along the parallel-wire transmission line(201) is given by Eq. (3) below.

$\begin{matrix}{U_{E} = {{\frac{1}{2}\frac{\rho_{\lambda}^{2}}{C}} = \frac{q^{2}}{sC}}} & {{Eq}.\mspace{14mu} (3)}\end{matrix}$

The total magnetic energy, u_(B), stored per unit length along theparallel-wire transmission line (201) is given by Eq. (4) below.

u _(B)=½lI ²  Eq. (4)

Accordingly, the Lagrangian of the parallel-wire transmission line (201)is given by Eq. (5) below.

$\begin{matrix}{\begin{matrix}{\mathcal{L} = {{U_{E} - U_{B}} = {\int{{dx}\left( {_{E} - _{B}} \right)}}}} \\{= {\int{{dx}\left\lbrack {{\frac{1}{2}\frac{\rho_{\lambda}^{2}}{C}} + \frac{q^{2}}{sC} - {\frac{1}{2}{lI}^{2}}} \right\rbrack}}} \\{= {\int{{dx}\left\lbrack {{\frac{- 1}{2}q\frac{q^{''}}{C}} + \frac{q^{2}}{sC} - {\frac{1}{2}l{\overset{.}{q}}^{2}}} \right\rbrack}}}\end{matrix}\quad} & {{Eq}.\mspace{14mu} (5)}\end{matrix}$

The generalized momentum π, the Euler-Lagrange equation of motion, andthe wave equation of the parallel-wire transmission line (201) are givenby Eq. (6), Eq. (7), and Eq. (8) below.

$\begin{matrix}{\pi = {{\partial_{\overset{.}{q}}} = {{- l}\overset{.}{q}}}} & {{Eq}.\mspace{14mu} (6)} \\{\overset{.}{\pi} = {{\partial_{q}} = {{{- l}\overset{¨}{q}} = {{- \frac{q^{''}}{c}} + {2\frac{q}{sC}}}}}} & {{Eq}.\mspace{14mu} (7)} \\{{- \overset{¨}{q}} = {{- \frac{q^{''}}{lc}} + {2\frac{q}{lsC}}}} & {{Eq}.\mspace{14mu} (8)}\end{matrix}$

Based on the wave equation Eq. (8), the dispersion relation for theparallel-wire transmission line (201) is given by Eq. (9a), Eq. (9b),and Eq. (9c) below.

$\begin{matrix}{v \equiv \frac{1}{\sqrt{lc}}} & {{Eq}.\mspace{14mu} \left( {9a} \right)} \\{\omega_{0} = \frac{1}{\sqrt{{lsC}/2}}} & {{Eq}.\mspace{14mu} \left( {9b} \right)} \\{\omega^{2} = {{v^{2}k^{2}} + \omega_{0}^{2}}} & {{Eq}.\mspace{14mu} \left( {9c} \right)}\end{matrix}$

In Eq. (9a), Eq. (9b), and Eq. (9c), ω represents an angular frequency,k represents a wave number, v represents an asymptotic wave velocity asdefined in Eq. (9a), and ω_(o) represents a cut off angular frequency asdefined in Eq. (9b). In particular, the cut off angular frequency ω_(o)is independent of the length, and varies logarithmically with the width,of the parallel-wire transmission line (201). In one or moreembodiments, one or more wire segments with associated capacitors of theparallel-wire transmission line (201) are detachable. Accordingly, theparallel-wire transmission line (201) may be re-configured, withoutsubstantially changing ω_(o), by the user to change the total lengthaccording to the dimension of the wireless power transfer area (101).

Based on Eq. (9c), FIG. 3A shows a plot of angular frequency, ω, versuswave number, k, to illustrate the dispersion relation for theparallel-wire transmission line (201). In addition, the phase velocity,v_(p), and group velocity, v_(g), are given in Eq. (10a) and Eq. (10b)below.

$\begin{matrix}{v_{p} = \frac{\omega}{k}} & {{Eq}.\mspace{14mu} \left( {10a} \right)} \\{v_{g} = \frac{\partial\omega}{\partial k}} & {{Eq}.\mspace{14mu} \left( {10b} \right)}\end{matrix}$

Note that as the wave number k asymptotically approaches 0, the phasevelocity v_(p) asymptotically approaches infinity, the group velocityv_(g) asymptotically approaches 0, and the angular frequency coasymptotically approaches ω₀.

FIG. 2B shows a schematic diagram of the parallel-wire transmission line(201) driven by the RF power source (108) in accordance with one or moreembodiments of the invention. In one or more embodiments, one or more ofthe modules and elements shown in FIG. 2B may be omitted, repeated,and/or substituted. Accordingly, embodiments of the invention should notbe considered limited to the specific arrangements of modules shown inFIG. 2B.

As shown in FIG. 2B, the parallel-wire transmission line (201) is drivenby the RF power source (108) connected via the terminal A (204 a) andterminal B (204 b). Further, the parallel-wire transmission line (201)is terminated by an electrically conducting connection (202) andoperating at the characteristic frequency ω_(o). In one or moreembodiments of the invention, the electrically conducting connection(202) may be substituted by a variable capacitor or other electroniccomponent, which may be used to fine tune the characteristic frequencyof the parallel-wire transmission line (201).

In one or more embodiments of the invention, the configuration of theparallel-wire transmission line (201) shown in FIG. 2B approximates thevariable form factor transmitter (102) depicted in FIG. 1A above.Similar to FIG. 1A, receiver devices (e.g., denoted as circular iconslabeled as A, B, C, D, E, and F) are disposed about the parallel-wiretransmission line (201) shown in FIG. 2B. The approximation isparticularly suitable for the scenario where the wireless power transferarea (101) has an elongated shape and where the string of distributedcapacitors of the variable form factor transmitter (102) is arrangedinto a pair of parallel lines according to the elongated shape of thewireless power transfer area (101). As described below, thecharacteristic frequency of the variable form factor transmitter (102)corresponds to ω_(o) described in reference to FIG. 2A above and issubstantially independent of the length, and varies logarithmically withthe width, of the parallel-wire transmission line (201).

In the configuration shown in FIG. 2B, the standing wave along theparallel-wire transmission line (201), as excited by the RF power source(108), has an infinite phase velocity. Therefore, the voltages andcurrents along the parallel-wire transmission line (201) are all inphase at different positions of the parallel-wire transmission line(201). In other words, the effective electrical length of theparallel-wire transmission line (201) equals zero regardless of thephysical length of the parallel-wire transmission line (201). In thescenario where there is no energy loss in the parallel-wire transmissionline (201), the input impedance of the parallel-wire transmission line(201) as presented to the RF power source (108) equals zero regardlessof the physical length of the parallel-wire transmission line (201). Inother words, the parallel-wire transmission line (201) is equivalent toan RLC circuit (not shown) resonant at ω_(o), regardless whether thephysical length of the parallel-wire transmission line (201) is muchshorter or much longer than the free-space wavelength (e.g., based onthe transmission medium of the wireless power transfer area (101)) ofthe driving frequency, i.e., ω_(o). Accordingly, the parallel-wiretransmission line (201) driven by the RF power source (108) andterminated by the electrically conducting connection (202) may be usedas a resonant power source for wireless power transfer to induceresonances of receiver devices that are placed in the vicinity of theparallel-wire transmission line (201). In particular, the resonantreceiver devices couple to the electric and/or magnetic fields generatedby the standing wave of the parallel-wire transmission line (201) andreceive power from the electric and/or magnetic fields.

In one or more embodiments, the resonant receiver devices receive powerfrom a near electromagnetic field of the parallel-wire transmission line(201). Even if the physical length of the parallel-wire transmissionline (201) is much longer than the free-space wavelength (e.g., based onthe transmission medium of the wireless power transfer area (101)) ofthe driving frequency, the power supplied from the RF power source (108)is substantially retained in the parallel-wire transmission line (201)for transferring to the nearby resonant receiver devices without beinglost to far field radiation. The quality factor of the parallel-wiretransmission line due to radiation loss depends only on the wireseparation and wire radius, not on the length.

FIG. 2C shows a variation of the parallel-wire transmission line (201)with distributed capacitance in which one of the conductor wires forms aconducting shield (203) that surrounds the other conductor wire,hereafter referred to as the shielded transmission line (201 a). Forexample, the conducting shield (203) may be substantially cylindrical.The shielded transmission line (201 a) shown in FIG. 2C operates by thesame principle as the parallel-wire transmission line (201) shown inFIG. 2B above, except the distributed capacitance is only placed on thecenter conductor. In some configurations, the center conductor may notbe concentric with the outer conductor (i.e., conducting shield 203).Further, the cross sections of the center conductor and outer conductor(i.e., conducting shield 203) may not be circular.

In one or more embodiments of the invention, the configuration of theshielded transmission line (201 a) shown in FIG. 2C approximates thevariable form factor transmitter (102) depicted in FIG. 1A above.Similar to FIG. 1A, receiver devices (e.g., denoted as circular iconslabeled as A, B, C, D, E, and F) are disposed about the parallel-wiretransmission line (201) shown in FIG. 2C. The approximation isparticularly suitable for the scenario where the wireless power transferarea (101) corresponds to the interior space within a conductiveenclosure, such as within a metal pipeline, an airframe of an airplaneor space shuttle, etc. The characteristic frequency of the variable formfactor transmitter (102), as shown in FIG. 2C, corresponds to ω_(o)described in reference to FIGS. 2A and 2B above and is substantiallyindependent of the length, and varies logarithmically with the diameter,of the conducting shield (203). The characteristic frequency of theshielded transmission line (201 a) shown in FIG. 2C is given by Eq.(11). Note that this differs from Eq. (9b) by a factor of √{square rootover (2)} due to the fact that only one of the conductor wires includesdistributed capacitors.

$\begin{matrix}{\omega_{0} = \frac{1}{\sqrt{lsC}}} & {{Eq}.\mspace{14mu} (11)}\end{matrix}$

FIG. 3B shows a plot of the quality factor, Q, of a parallel-wiretransmission line (e.g., shown in FIG. 2A or FIG. 2B) of arbitrarylength, consisting of 14 AWG copper wire, driven at 6.78 MHz, as afunction of the separation, d, (between the two wires) divided by thefree-space wavelength λ. For wire separations large relative to thefree-space wavelength, the Q is suppressed due to radiation loss.However, for wire separations small compared to the free-spacewavelength, the radiation is suppressed and the loss is dominated byohmic losses in the copper wire.

Note that the shielded transmission line (201) has no radiative loss dueto the fact that the conducting shield (203) completely encloses theinternal electromagnetic fields.

In contrast, while a conducting wire loop driven by the RF power source(108), described in reference to FIG. 2D below, may also transfer powerto resonant receiver devices in the vicinity, the efficiency of thepower transfer is decreased due to far field radiation as the dimensionof the conducting wire loop increases to approach or exceed thefree-space wavelength of the driving frequency. FIG. 3C shows a plot ofthe quality factor, Q, of a circular loop consisting of 14 AWG copperwire, driven at 6.78 MHz, as a function of the loop radius a divided bythe free-space wavelength λ. Note that the Q becomes low, and thereforethe efficiency of wireless power transfer is suppressed, as the loopradius becomes large relative to the free-space wavelength.

FIG. 2D shows a schematic diagram of a wire loop (204) havingdistributed capacitors and driven by the RF power source (108) inaccordance with one or more embodiments of the invention. In one or moreembodiments, one or more of the modules and elements shown in FIG. 2Dmay be omitted, repeated, and/or substituted. Accordingly, embodimentsof the invention should not be considered limited to the specificarrangements of modules shown in FIG. 2D.

In one or more embodiments, the wire loop (204) has a circular loopradius, a, and a wire radius (corresponding to a gauge of the wire), b,(not shown) and is composed of wire segments of length s joined by anumber of capacitors, C. In one or more embodiments of the invention,the configuration of the wire loop (204) shown in FIG. 2D approximatesthe variable form factor transmitter (102) depicted in FIG. 1A above.The approximation is particularly suitable for the scenario where aparticular shape of the wireless power transfer area (101) matches thecircular form factor of the variable form factor transmitter (102). Asis described below, the characteristic frequency of the variable formfactor transmitter (102) corresponds to a resonant frequency ω_(o) ofthe wire loop (204) and is substantially independent of the width and/orlength (i.e., form factor) of the wire loop (204).

The inductance, L, the total capacitance, C_(tot), and the resonantangular frequency, ω_(o), of the wire loop (204) are given by Eq. (12a),Eq. (12b), and Eq. (12c) below.

$\begin{matrix}{L = {\mu \; {a\left\lbrack {{\ln \left( \frac{8a}{b} \right)} - 2} \right\rbrack}}} & {{Eq}.\mspace{14mu} \left( {12a} \right)} \\{C_{tot} = {\frac{C}{N} = \frac{C}{\left( {2\pi \; {a/s}} \right)}}} & {{Eq}.\mspace{14mu} \left( {12b} \right)} \\\begin{matrix}{\omega_{0}^{2} = {\frac{1}{{LC}_{tot}} = \frac{1}{\mu \; {{a\left\lbrack {{\ln \left( \frac{8a}{b} \right)} - 2} \right\rbrack} \cdot \frac{s}{2\; \pi \; a} \cdot C}}}} \\{= \frac{1}{\frac{\mu}{2\; \pi}{{sC}\left\lbrack {{\ln \left( \frac{8a}{b} \right)} - 2} \right\rbrack}}}\end{matrix} & {{Eq}.\mspace{14mu} \left( {12c} \right)}\end{matrix}$

In Eq. (12a), Eq. (12b), and Eq. (12c), N denotes the number of wiresegments or capacitors, C, in the wire loop (204) and μ denotes theelectro-magnetic permeability of the transmission medium in the wirelesspower transfer area (101). In one or more embodiments, the resonantangular frequency, ω_(o), depends only weakly on the radius, a, of thewire loop (204) or the wire radius, b. In one or more embodiments, oneor more wire segments with associated capacitors of the wire loop (204)are detachable. Accordingly, the wire loop (204) may be reconfigured,without substantially changing the resonant angular frequency ω_(o), bythe user to change the loop radius, a, according to the dimensions ofthe wireless power transfer area (101).

Unlike the parallel-wire transmission line (201) shown in FIG. 2A above,the wire loop (204) becomes an efficient far field radiator as theradius, a, becomes comparable to or exceeds the free-space wavelength(e.g., based on the transmission medium of the wireless power transferarea (101)) of the driving frequency, i.e., ω_(o). The radiationresistance (i.e., effective series resistance due to far fieldradiation) R_(rad) of a closed loop of wire carrying a uniform currentis given by the double integral over the wire path shown Eq. (13a)below.

$\begin{matrix}{R_{rad} = {\frac{{\zeta\kappa}^{2}}{4\; \pi}{\int{{{dr}_{1} \cdot {dr}_{2}}\frac{\sin \left( {\kappa {{r_{1} - r_{2}}}} \right)}{\kappa {{r_{1} - r_{2}}}}}}}} & {{Eq}.\mspace{14mu} \left( {13a} \right)} \\{\zeta = \sqrt{\frac{\mu}{\epsilon}}} & {{Eq}.\mspace{14mu} \left( {13b} \right)} \\{{\kappa \equiv \frac{\omega}{c}} = \frac{2\pi}{\lambda}} & {{Eq}.\mspace{14mu} \left( {13c} \right)}\end{matrix}$

In Eq. (13a), based on the transmission medium of the wireless powertransfer area (101), ζ is the impedance of free space, and κ is thefree-space wavenumber. Based on Eq. (13a) applied to the wire loop(204), FIG. 3D shows a plot of radiation resistance divided by theimpedance of free space as a function of radius divided by wavelength.As can be seen from FIG. 3D, the radiation resistance has the asymptoticforms for large and small loop radius given in Eq. (14) below.

$\begin{matrix}{{{\frac{R_{rad}}{\zeta} \approx {\frac{8\; \pi^{5}}{3}\left( \frac{a}{\lambda} \right)^{4}}},{a\lambda}}{{\frac{R_{rad}}{\zeta} \approx {\pi^{2}\left( \frac{a}{\lambda} \right)}},{a\lambda}}} & {{Eq}.\mspace{14mu} (14)}\end{matrix}$

The quality factor, Q, of the loop due to radiation is equal to theratio of the inductive reactance, ω_(o)L, divided by the total seriesresistance, R, which includes the radiation resistance, R_(rad). As theradiation resistance increases, the quality factor decreases, causingthe efficiency of the wireless power transfer to decrease.

For the circular wire loop (204) shown in FIG. 2D, Eq. (12c) applieswhere ω₀=√{square root over (2π/(μsC(ln(8a/b)−2)))} with a being theloop radius and b being the wire radius. For the parallel-wiretransmission line (201) shown in FIG. 2B, Eq. (9b) applies and it can beshown that ω₀=√{square root over (2π/(μsCln(d/b)−2)))}, with d being thewidth of the parallel-wire transmission line and b being the wireradius. The characteristic frequencies, ω₀, have similar values for bothcircular loop and parallel-wire configurations if ln(a/b)≈ln(d/b). Inthis manner, a single variable form factor transmitter (102) may bemanufactured for use in both elongated-shaped service area andcircular-shaped service area based on the user adapted elongated formfactor or circular form factor. In other words, based on the wirediameter, b, used to manufacture the variable form factor transmitter(102), the user may select the loop radius, a, and the parallel-wiretransmission line width, d, such that ln(a/b)≈ln(d/b). In this manner,one single variable form factor transmitter manufactured in the factorycan be configured into either a parallel-wire form factor depicted inFIG. 2B or a circular form factor depicted in FIG. 2D to supply power tothe same set of receiving devices that are tuned to the particularresonant frequency, ω₀.

FIG. 2E shows a schematic diagram of a rectangular loop (206) havingdistributed capacitors and driven by the RF power source (108) inaccordance with one or more embodiments of the invention. In one or moreembodiments of the invention, the configuration of the rectangular loop(206) approximates the variable form factor transmitter (102) depictedin FIG. 1A above. Similar to FIG. 1A, receiver devices (e.g., denoted ascircular icons labeled as A, B, C, D, E, and F) are disposed about therectangular loop (206) shown in FIG. 2E. For example, the rectangularloop (206) may correspond to the parallel-wire transmission line (201)shown in FIG. 2B that has been adapted by a user to fit arectangular-shaped wireless power transfer area. In another example, therectangular loop (206) may correspond to the wire loop (204) shown inFIG. 2D that has been adapted by a user to fit a rectangular-shapedwireless power transfer area. As shown in FIG. 2E, the rectangular loop(206) is driven by the RF power source (108) using a transformercoupling scheme. In particular, the transformer (206 a) includes acapacitor C₂ in parallel to the primary coil L₁ and a capacitor C₁ inparallel to the secondary coil L₁. In addition, the electricallyconducting connection (202) shown in FIG. 2B is substituted by acapacitor C₂. The capacitance values of the capacitors C₁, C₂, and C₃may be adjusted in the factory and/or by the user for impedance matchingbetween the power source (108) and the rectangular loop (206) and fortuning the resonant frequency of the rectangular loop (206).

FIG. 2F shows a schematic diagram of connecting the power source (108)using a capacitive coupling scheme. In particular, the power source(108) is connected to a distributed capacitor string (207), via acoaxial cable (208) and a twisted pair (209), at opposite terminals of atuning capacitor C₁. The value of the tuning capacitor C₁ may beadjusted in the factory or by the user to provide a proper impedancematch to both the RF power source (108) and the coaxial cable (208). Byattaching the shield of the coaxial cable (208) to a voltage node of thedistributed-capacitor string (207), the shield of the coaxial cable(208) is maintained at ground potential.

In one or more embodiments, the distributed capacitor string (207) maycorrespond to a portion of the parallel-wire transmission line (201)shown in FIGS. 2B and 2C, a portion of the wire loop (204) shown in FIG.2D, or a portion of the rectangular loop (206) shown in FIG. 2E. Thevoltage magnitude relative to ground (210) induced by the power source(108), is shown as a function of the position along the distributedcapacitor string (207).

FIG. 2G shows a schematic diagram for connecting the power source (108)to the variable form factor transmitter using an alternative capacitivecoupling scheme. As shown in FIG. 2G, a resonant balun (211) is used toconnect the power source (108) to the tuning capacitor, C₁.

FIG. 3E is a plot of inductance as a function of the aspect ratio(represented by width/half-perimeter) of a rectangular loop (e.g.,rectangular loop (206) depicted in FIG. 2E above), made from 83 feet of14 AWG wire, and driven at 6.78 MHz. The rectangular loop with the rangeof aspect ratios shown in FIG. 3E represents various shapes the wireloop (204) shown in FIG. 2D may be adapted by the user to fit anywireless power transfer area. The plot shows the inductance of therectangular loop as the perimeter (i.e., corresponding to thecircumference of the wire loop (204)) is held fixed but the aspect ratiois varied. As can be seen from the plot, the inductance varies less than20% as the aspect ratio is varied over a wide range between 0.05 and0.95. Accordingly, the characteristic frequency of the wire loop (204)varies less than 10% while being adapted into a rectangular loop over awide range of aspect ratios. This demonstrates the relativeinsensitivity of the resonant frequency of the loop with distributedcapacitance to variations in the adapted form factor.

Referring back to the discussion of FIG. 1A, in one or more embodimentsof the invention, the system (100) provides wireless power transferacross the wireless power transfer area (101) based on the ISM band. Inthe scenario where the variable form factor transmitter (102) isapproximated by the parallel-wire transmission line (201) shown in FIG.2A, 2B, or 2C, the values of the wire segment length s, the inductanceper unit length l, and the capacitor C may be chosen in the factory,based on Eq. (9b), to maintain the resonant angular frequency ω_(o) ofthe parallel-wire transmission line (201) equal to the angular frequencyof the RF power source, which may be within the type A frequency range(i.e., 6.765 MHz-6.795 MHz) defined in the ITU Radio Regulations Article5, footnote 5.138.

In the scenario where the variable form factor transmitter (102) isapproximated by the wire loop (204) shown in FIG. 2D, the values of thewire segment length, s, and the capacitor, C, may be chosen in thefactory, based on Eq. (12c), to maintain the resonant angular frequencyω_(o) of the wire loop (204) equal to the angular frequency of the RFpower source, which may be within the type A frequency range (i.e.,6.765 MHz-6.795 MHz) defined in the ITU Radio Regulations Article 5,footnote 5.138.

In one or more embodiments of the invention, the aforementionedmanufacturing tolerance is controlled such that the resultingcapacitance range, length range, and inductance range do not cause theresonant angular frequency ω_(o) to deviate from the type A frequencyrange (i.e., 6.765 MHz-6.795 MHz). In addition for both scenariosdescribed above, approximation error exists due to physical differencebetween the user adapted form factor of the variable form factortransmitter (102) and the simplified form factor of the parallel-wiretransmission line (201) or the wire loop (204). In one or moreembodiments of the invention, to compensate for the aforementionedmanufacturing tolerance and the approximation error, the input impedanceand the characteristic frequency of the variable form factor transmitter(102) may be adjustable in the factory as well as by the user.

Further to the discussion of FIG. 1A above, FIG. 2H shows schematicdiagrams of an equivalent circuit A (205 a) and an equivalent circuit B(205 b) of the variable form factor transmitter (102). For optimal powertransfer from the power source (108), the input impedance of thevariable form factor transmitter (102) is matched to the outputimpedance (represented by the resistor R_(L)) of the power source (108).The resistor R is an effective series resistance representing allsources of loss (e.g., ohmic loss, radiation loss, dielectric loss,etc.) of the variable form factor transmitter (102). The variablecapacitor C₁ determines the apparent input impedance of the variableform factor transmitter (102) at its resonant frequency, while thevariable capacitor C₂ sets the resonant frequency.

The equivalent circuit B (205 b) corresponds to a simplified form of theequivalent circuit A (205 a) where C₂, C₃, and L have been combined intoa single reactance, χ. The input impedance of the variable form factortransmitter (102) is equal to R_(L) when C₁ has the value given by Eq.(15).

$\begin{matrix}{C_{1} = {\frac{1}{\omega_{o}R_{L}}\sqrt{\frac{R_{l}}{R} - 1}}} & {{Eq}.\mspace{14mu} (15)}\end{matrix}$

For the case where R_(L)<R, the transformer coupling scheme shown inFIG. 2E may be used. For the case where R_(L)≥R, the capacitive couplingscheme shown in FIG. 2F may be used.

FIG. 2J shows example constructions of the variable form factortransmitter (102), depicted based on the legend (221), in accordancewith one or more embodiments. In one or more embodiments, one or more ofthe modules and elements shown in FIG. 2J may be omitted, repeated,and/or substituted. Accordingly, embodiments of the invention should notbe considered limited to the specific arrangements of modules shown inFIG. 2J.

As shown in FIG. 2J, the distributed capacitor string A (210 a) anddistributed capacitor string B (210 b) are two example constructions ofthe variable form factor transmitter (102) depicted in FIGS. 1.1 and 1.2above. Accordingly, the wire loop (204) and the rectangular loop (206),respectively depicted in FIGS. 2D and 2E above, may be based on thedistributed capacitor string A (210 a), the distributed capacitor stringB (210 b), or a combination thereof. In particular, the distributedcapacitor string A (210 a) and distributed capacitor string B (210 b)show two example constructions corresponding to a cross-sectional viewof the wire loop (204) or the rectangular loop (206). Specifically, thecross-sectional view includes a cross-section of consecutive capacitorsand wire-segments in the wire loop (204) or the rectangular loop (206).

The distributed capacitor string A (210 a) includes capacitorsconstructed from conductive strips (230 a), (230 b), (230 c), (230 d),(230 e), (230 f), etc. attached to two opposing surfaces (i.e., surfaceA (220 a), surface B (220 b)) of a dielectric material sheet.Specifically, a sheet has a three-dimensional (3D) form factor with themajority (e.g., greater than 90%) of surface area occupied by the twoopposing surfaces. In other words, the thickness (i.e., distance betweenthe two opposing surfaces) of the sheet is substantially less than eachdimension of the two opposing surfaces. In this context, the 3D formfactor of the sheet may be represented as a two-dimensional (2D) formfactor with the thickness along a third dimension perpendicular to asurface (e.g., surface A (220 a), surface B (220 b)) of the 2D formfactor. A conductive strip is a sheet of conductive material that isattached to, and has a substantially smaller (e.g., less than 10%) areathan, the dielectric material sheet. The distributed capacitor string A(210 a) is depicted in a cross-sectional view showing cross-sections ofthe conductive strips and the dielectric material sheet. In particular,the cross-sectional view cuts across the surface A (220 a) and surface B(220 b) along the third dimension to show the thicknesses of theconductive strips and the dielectric material sheet.

In one or more embodiments, one or more of the conductive strips (230a), (230 b), (230 c), (230 d), (230 e), (230 f), etc. are printed on thesurface A (220 a) and/or surface B (220 b) using conductive ink, paste,paint, or other conductive coating material. In one or more embodiments,one or more of the conductive strips (230 a), (230 b), (230 c), (230 d),(230 e), (230 f), etc. are formed by selectively etching one or moreconductive films laminated with the dielectric material sheet. Forexample, the conductive film(s) and the dielectric material sheet may belaminated together by heat, pressure, adhesive, welding, or othersuitable method.

For example, the capacitor (211) includes overlapping portions of theconductive strip (230 a) and conductive strip (230 e), respectivelyattached to the surface A (220 a) and surface B (220 b), that areseparated by a thickness d of the dielectric material sheet. Theoverlapping portions of the conductive strip (230 a) and conductivestrip (230 e) form two electrodes in a parallel-plate configuration ofthe capacitor (211). Similarly, the capacitor (213) includes overlappingportions of the conductive strip (230 b) and conductive strip (230 e),respectively attached to the surface A (220 a) and surface B (220 b),that are separated by the thickness d of the dielectric material sheet.The overlapping portions of the conductive strip (230 b) and conductivestrip (230 e) form two electrodes in the parallel-plate configuration ofthe capacitor (213). The overlapping portions of two adjacent conductivestrips are referred to as the overlap region having a distance x.Further, each of the conductive strips (230 a), (230 b), (230 c), (230d), (230 e), (230 f), etc. acts as an inductive segment that connectstwo adjacent capacitors in the distributed capacitor string A (210 a).For example, the conductive strip (230 e) acts as, or otherwiseimplements, the inductive segment (212) to connect the capacitor (211)and capacitor (213) in series. Accordingly, the capacitor (211) and theinductive segment (212) form one of the multiple capacitor-wire segmentsof the distributed capacitor string A (210 a). Similarly, the capacitor(213) and the inductive segment (214) form another one of the multiplecapacitor-wire segments of the distributed capacitor string A (210 a).In the distributed capacitor string A (210 a), the surface of thedielectric material sheet, where the inductive segments are attached,alternates between the surface A (210 a) and surface B (210 b). Forexample, the inductive segment (212) and one electrode of the capacitor(213) are integrated as a single conductive strip (230 e) attached tothe surface A (210 a), while the inductive segment (214) and the otherelectrode of the capacitor (213) are integrated as a single conductivestrip (230 b) attached to the opposing surface B (210 b). In thiscontext, each capacitor-wire segment shown in the distributed capacitorstring A (210 a) is a first type of integrated capacitor-wire segment.As used herein, the integrated capacitor-wire segment is a capacitor andan inductive segment that are connected in series where the inductivesegment and one electrode of the capacitor are integrated into a singleconductive strip.

Further as shown in FIG. 2J, the distributed capacitor string B (210 b)includes capacitors constructed from conductive strips (230 g), (230 h),(230 j), (230 k), (230 m), (230 n), (230 p), etc. attached to twoopposing surfaces (i.e., surface C (220 c), surface D (220 d)) of adielectric material sheet. Similar to the distributed capacitor string A(210 a), the distributed capacitor string B (210 b) may be constructedby printing, lamination, etching, or combinations thereof. For example,the capacitor (215) includes overlapping portions of the conductivestrip (230 g) and conductive strip (230 m), respectively attached to thesurface C (220 c) and surface D (220 d), that are separated by thethickness d of the dielectric material sheet. The overlapping portionsof the conductive strip (230 g) and conductive strip (230 m) form twoelectrodes in the parallel-plate configuration of the capacitor (215).The capacitor (216) includes overlapping portions of the conductivestrip (230 h) and conductive strip (230 m), respectively attached to thesurface C (220 c) and surface D (220 d), that are separated by thethickness d of the dielectric material sheet. The overlapping portionsof the conductive strip (230 h) and conductive strip (230 m) form twoelectrodes in the parallel-plate configuration of the capacitor (216).The capacitor (215) and capacitor (216) are connected together in seriesat the conductive strip (230 m) to form a combined capacitor (222) thatis itself connected between the conductive strip (230 g) and conductivestrip (230 h). Similarly, the combined capacitor (223), including twocapacitors connected in series, is connected between the conductivestrip (230 h) and conductive strip (230 j). Further, each of theconductive strips (230 g), (230 h), (230 j), etc. acts as an inductivesegment that connects two adjacent combined capacitors in thedistributed capacitor string B (210 b). For example, the conductivestrip (230 h) acts as, or otherwise implements, the inductive segment(218) to connect the combined capacitor (222) and combined capacitor(223) in series. Accordingly, the combined capacitor (222) and theinductive segment (218) form one of the multiple capacitor-wire segmentsof the distributed capacitor string B (210 b). Similarly, the combinedcapacitor (223) and the inductive segment (219) form another one of themultiple capacitor-wire segments of the distributed capacitor string B(210 b). In the distributed capacitor string B (210 b), the inductivesegments (215), (218), (219), etc. are attached to a single surface(i.e., surface C (220 c)) of the dielectric material sheet. For example,the inductive segment (218) and one electrode of the combined capacitor(223) are integrated as a single conductive strip (230 h) attached tothe surface C (210 c), while the inductive segment (219) and the otherelectrode of the combined capacitor (223) are integrated as a singleconductive strip (230 j) attached to the same surface C (210 c). In thiscontext, each capacitor-wire segment shown in the distributed capacitorstring B (210B) is a second type of integrated capacitor-wire segment.

As noted above, the wire loop (204) and the rectangular loop (206),respectively depicted in FIGS. 2D and 2E above, may be based on thedistributed capacitor string A (210 a), the distributed capacitor stringB (210 b), or a combination thereof. In other words, the first type ofintegrated capacitor-wire segment(s) in the distributed capacitor stringA (210 a) and/or the second type of integrated capacitor-wire segment(s)in the distributed capacitor string B (210 b) may be included in thewire loop (204) and/or the rectangular loop (206), respectively depictedin FIGS. 2D and 2E above. Although a specific number of integratedcapacitor-wire segments are shown in the distributed capacitor string A(210 a) and the distributed capacitor string B (210 b) above, the wireloop (204) and/or the rectangular loop (206) may also include morenumber of integrated capacitor-wire segments of either type, or lessnumber of integrated capacitor-wire segments of either type, than whatis shown in the distributed capacitor string A (210 a) and thedistributed capacitor string B (210 b). In one or more embodiments,either type or both types of integrated capacitor-wire segments may becombined with other forms of capacitor-wire segments (e.g., based ondiscrete capacitor(s) and inductor(s)) to form the wire loop (204)and/or the rectangular loop (206), respectively depicted in FIGS. 2D and2E above.

FIG. 2K shows an example construction of the variable form factortransmitter (102), depicted based on the legend (221), in accordancewith one or more embodiments. In one or more embodiments, one or more ofthe modules and elements shown in FIG. 2K may be omitted, repeated,and/or substituted. Accordingly, embodiments of the invention should notbe considered limited to the specific arrangements of modules shown inFIG. 2K.

As shown in FIG. 2K, the power transmitter (250), depicted based on thelegend (221), shows an example construction corresponding to athree-dimensional (3D) view of FIG. 2E above. The rectangular loop (206)of the power transmitter (250) is based on the distributed capacitorstring A (210 a) described in reference to FIG. 2J above. Specifically,a portion (224) of the rectangular loop (206) corresponds to a 3D viewof the distributed capacitor string A (210 a) depicted in FIG. 2J. Inother words, the distributed capacitor string A (210 a) depicted in FIG.2J corresponds to a cross section (designated by the double arrowed anddashed line) of the portion (224). The thicknesses of the conductivestrips and the dielectric material sheet in the distributed capacitorstring A (210 a) are omitted in the 3D view for clarity of showing theaforementioned 2D form factors.

RF characteristics of the rectangular loop (206) is described below,where the width, length, and number of overlap regions of therectangular loop (206) are denoted as a, b, and n, respectively. Theoverlap area A between two adjacent conductive strips may be computedusing Eq. (16), where the conductive strip width, conductive striplength, and overlapping region distance are denoted as w, l, and x,respectively.

A=w×x  Eq. (16)

The capacitance of each overlap region may be computed using Eq. (17),where the dielectric constant and the thickness of the dielectricmaterial sheet, are denoted as ε and d, respectively.

$\begin{matrix}{C = \frac{ɛ\; A}{d}} & {{Eq}.\mspace{14mu} (17)}\end{matrix}$

The total capacitance of the rectangular loop (206) may be computedusing Eq. (18).

$\begin{matrix}{C_{tot} = \frac{C}{n}} & {{Eq}.\mspace{14mu} (18)}\end{matrix}$

The total inductance of the rectangular loop (206) may be computed usingEq. (19) and Eq. (20).

$\begin{matrix}\left. {L = {\frac{\mu}{\pi}\left\lbrack {{a\mspace{11mu} \ln \frac{2\; a}{\rho}} + {b\mspace{11mu} \ln \frac{2\; b}{\rho}} + {2\sqrt{a^{2} + b^{2}}} - {a\mspace{11mu} \sinh^{- 1}\frac{a}{b}} - \; {b\; \sinh^{- 1}\frac{b}{a}} - {2\left( {a + b} \right)}} \right)}} \right\rbrack & {{Eq}.\mspace{14mu} (19)} \\{\mspace{79mu} {\rho = \frac{\omega}{4}}} & {{Eq}.\mspace{14mu} (20)}\end{matrix}$

The resonant frequency ω_(o) of the rectangular loop (206) may becomputed using Eq. (21).

$\begin{matrix}{\omega_{o} = \frac{1}{\sqrt{{LC}_{tot}}}} & {{Eq}.\mspace{14mu} (21)}\end{matrix}$

Additional relationships between the resonant frequency and otherparameters of the rectangular loop (206) include Eq. (22), Eq. (23), andEl. (24).

$\begin{matrix}{C_{tot} = \frac{1}{L\; \omega_{o}^{2}}} & {{Eq}.\mspace{14mu} (22)} \\{\frac{ɛ\; A}{d} = \frac{n}{L\; \omega_{o}^{2}}} & {{Eq}.\mspace{14mu} (23)} \\{\frac{A}{d} = \frac{n}{ɛ\; L\; \omega_{o}^{2}}} & {{Eq}.\mspace{14mu} (24)}\end{matrix}$

TABLE 2 lists four examples of the RF characteristics of the rectangularloop (206) based on the equations above.

TABLE 2 d (nm) A (mm²) n ε (F/m) L (H) ω_(o) ² (Hz²) 894.5 63.5 20 8.854E−12 1.753 E−6 1.8148 E15 100.3 6.35 20 8.854 E−12 1.753 E−6 1.8148 E15767.3 63.5 10 8.854 E−12 1.753 E−6 1.8148 E15 8.945 63.5 20 8.854 E−121.753 E−6 1.8148 E13

In one or more embodiments, the power transmitter (250) is configuredbased on a pre-determined wireless power transfer area. For example, thepre-determined wireless power transfer area may be a table top surfacewhere one or more mobile receiver devices (e.g., mobile phone) areplaced to receive wireless power transfer. The rectangular loop (206)may be movably or permanently disposed along a path based on the tabletop surface. For example, the path may be the edges of the table topsurface, on top of or beneath the table top surface, on a fixture orceiling above the table top surface, on or embedded in the floor belowthe table top surface, etc. The power source (108) is connected to thecapacitor-wire segments via the terminal A (202 a) and terminal B (202b), and may be plugged into a power outlet on a wall near the table topsurface. The dielectric material sheet (225) encompasses at least aportion of the path to implement the capacitor(s) of the rectangularloop (206) and to provide mechanical support for the rectangular loop(206).

In another example, the pre-determined wireless power transfer area maybe a space adjacent to a window where one or more receiver devices(e.g., mobile phone) are disposed about the space to receive wirelesspower transfer. The rectangular loop (206) may be movably or permanentlydisposed along a path based on the window. For example, the path may bethe edges of the window frame, in front of or behind the window glasssurface, embedded in the window glass or the window frame, etc. Thepower source (108) may be plugged into a power outlet on a wall wherethe window is mounted, or wired to the power outlet behind the surfaceof the wall.

In one or more embodiments, one or more of the dielectric materialsheet, conductive strips, and/or integrated capacitor-wire segments maybe rigid or pliable, transparent, translucent, or opaque depending onrespective thicknesses and/or compositions. Although 2D form factors ofthe dielectric material sheet, conductive strips, and/or integratedcapacitor-wire segments are shown as rectangular shapes in FIG. 2K,different 2D form factors (e.g., polygonal, circular, oval, elliptical,spiral, etc. shapes or combinations thereof) than what is shown may alsobe exhibited by the dielectric material sheet, conductive strips, and/orintegrated capacitor-wire segments. Although the conductive strips inthe rectangular loop (206) follows a path outlining a rectangular shapein FIG. 2K, the conductive strips in the rectangular loop (206) may alsofollow a different path outlining a different shape that turns therectangular loop (206) into a loop with a different shape, such as apolygonal, circular, oval, elliptical, spiral, etc. loop or combinationsthereof. Although a specific number of conductive strips and/orintegrated capacitor-wire segments in the rectangular loop (206) areshown in FIG. 2K, the power transmitter (250) may also include morenumber of integrated capacitor-wire segments, or less number ofintegrated capacitor-wire segments than what is shown.

FIG. 2L shows an example construction of the variable form factortransmitter (102) in accordance with one or more embodiments. In one ormore embodiments, one or more of the modules and elements shown in FIG.2L may be omitted, repeated, and/or substituted. Accordingly,embodiments of the invention should not be considered limited to thespecific arrangements of modules shown in FIG. 2L.

As shown in FIG. 2L, the power transmitter (260), depicted based on thelegend (231), shows an example construction corresponding to a top viewof the power transmitter (250) depicted in FIG. 2K above. In particular,the top view has a viewing direction along the aforementioned thirddimension perpendicular to the surface (e.g., surface A (220 a), surfaceB (220 b)) of the power transmitter (250). The power transmitter (260)includes a rectangular loop (206 a) implemented using the dielectricmaterial sheet (251) and connected to the power source (108). Therectangular loop (206 a) and dielectric material sheet (251) arevariations of the rectangular loop (206) and dielectric material sheet(225) depicted in FIG. 2K above. For, example, the rectangular loop (206a) and the rectangular loop (206) have different number ofcapacitor-wire segments. Further, the dielectric material sheet (251)includes an opening (233) where the dielectric material is cut out fromthe dielectric material sheet (251).

In one or more embodiments, the power source (108) is implemented usingat least a flexible circuit having a thin insulating polymer film withconductive circuit patterns and electronic chips affixed thereto. Forexample, the flexible circuit may be attached to and/or mechanicallysupported by the dielectric material sheet (251). Example details of aportion of the power transmitter (260) containing the power source (108)is shown in FIG. 2M below based on the legend (241).

As shown in FIG. 2M, the power source (108) includes a flexible circuit(108 a) connected to several conductive strips attached to a surface ofthe dielectric material sheet (251). The conductive strips include aspiral A (209 a), a spiral B (209 b), a spiral C (209 c), and a spiral D(209 d). One end of the spiral A (209 a) is designated as the terminal A(204 a), one end of the spiral B (209 b) is designated as the terminal B(204 b), one end of the spiral C (209 c) is designated as the terminal C(204 c), and one end of the spiral D (209 d) is designated as theterminal D (204 d). The other ends of the spiral A (209 a) and spiral B(209 b) are connected together using the conductive bridge A (209 d) toimplement a secondary winding of an isolation transformer contained inthe power source (108). The other ends of the spiral C (209 c) andspiral D (209 d) are connected together using the conductive bridge B(209 e) to implement a primary winding of the isolation transformer. Theconductive bridge A (209 d) and conductive bridge B (209 e) may beimplemented using insulated conductive wires or other electricalconnection means. The primary and secondary windings intertwine witheach other to provide an inductance coupling effect of the isolationtransformer. In addition, certain capacitors (designated as C1 and C2)contained in the power source (108) may be connected to the terminal A(204 a), terminal B (204 b), terminal C (204 c), and terminal D (204 d).The capacitors C1 and C2 may be discrete capacitors soldered to theterminals or capacitors implemented using additional conductive stripsattached to the two opposing surfaces of the dielectric material sheet(251). For example, the isolation transformer and the capacitors C1 andC2 may be part of or related to an impedance matching circuit tosubstantially match a pre-determined output impedance of the powersource (108) to the string of distributed capacitors in the rectangularloop (206 a) depicted in FIG. 2L.

FIG. 2N shows an application example of a wireless power transfer area,based on the power transmitter (250) depicted in FIG. 2K above, inaccordance with one or more embodiments. In one or more embodiments, oneor more of the modules and elements shown in FIG. 2N may be omitted,repeated, and/or substituted. Accordingly, embodiments of the inventionshould not be considered limited to the specific arrangements of modulesshown in FIG. 2N.

As shown in FIG. 2N, the wireless power transfer area includes the tabletop (600) where the rectangular loop (206) of the power transmitter(250) follows the edges of the table top (600). The dielectric materialsheet (225) is laid on top of the table top (600) where the thickness isomitted for clarity. The power cord and plug of the power source (108)are also omitted. The receiver device A (500 a) and receiver devices B(500 b) receive wireless power transfer from the power transmitter (250)to light up a string of decorative light emitting diodes (LEDs) attachedto the bottoms of glasses. Examples of the receiver device A (500 a) andreceiver devices B (500 b) are described in reference to FIGS. 5A, 5B,5D, and 5E below. In addition, the receiver device C (500 c) is acommercially available product that receives wireless power transferfrom the power transmitter (250) to charge the battery of a mobiledevice (500), such as a mobile phone, tablet computer, notebookcomputer, etc. TABLE 3 shows input power and input current for fourexample loading scenarios of the power transmitter (250).

TABLE 3 Input power (W) Input current (A) Number of Load 23.23 1.96 023.80 2.01 1 LED 24.82 2.10 2 LED 25.63 2.17 3 LED 28.38 2.17 3 LED +battery charger

Similar to the distributed capacitor string A (210 a) and distributedcapacitor string B (210 b) depicted in FIG. 2J above, FIG. 2P showsadditional constructions of the variable form factor transmitter inaccordance with one or more embodiments. In particular, overlappingportions of the two conducting films correspond to capacitors whilenon-overlapping portions of either conducting film correspond toinductors. In one or more embodiments, one or more of the modules andelements shown in FIG. 2P may be omitted, repeated, and/or substituted.Accordingly, embodiments of the invention should not be consideredlimited to the specific arrangements of modules shown in FIG. 2P.

As shown in FIG. 2P, the distributed capacitor string C (210 c),distributed capacitor string D (210 d), and distributed capacitor stringE (210 e) are three additional example constructions of the variableform factor transmitter (102) depicted in FIGS. 1.1 and 1.2 above.According to the legend (300), the dielectric is a layer which separatestwo layers of conducting film. The mechanical support is not provided bythe dielectric, but rather by a separate and distinct insulatingmechanical substrate that is shown below both the conducting film layersand the dielectric layer. The dielectric may include an oxide layergrown on the surface of one of the layers of conducting film, which maybe a metallic conductor. The dielectric layer may cover the entire uppersurface of the lower conducting layer, as shown in the distributedcapacitor string C (210 c), or may only cover the region of overlap, asshown in the distributed capacitor string D (210 d).

Alternatively, as shown in the distributed capacitor string E (210 e),the dielectric may consist of a thin film of insulating material towhich both the upper and lower conducting films adhere. However, thedielectric may be too thin to provide adequate mechanical support, inwhich case all three layers may be superposed above an additionalinsulating layer, which provides mechanical support of the dielectricand upper/lower conducting films.

FIG. 4A shows a schematic diagram of an example RF power source inaccordance with one or more embodiments of the invention. In particular,the example RF power source (108) shown in FIG. 4A may operate based onthe ISM band as the power source (108) depicted in FIGS. 1A, 1C, 2B, 2C,2D, 2K, 2L, and 2M above. Specifically, the example RF power source(108) shown in FIG. 4A includes the terminal A (204 a) and terminal B(204 b) that correspond to the two terminals of the power source (108)depicted in FIGS. 1A, 1C, 2B, 2C, 2D, 2K, 2L, and 2M above. Theschematic diagram includes capacitors, inductors, and resistors ofvarious RLC circuit components and commercial part numbers of variousintegrated circuit components. In particular, the inductors designatedas L1 and L2 correspond to the primary conductive winding and secondaryconductive winding shown in FIG. 2M above. The capacitors designated asC1 and C2 correspond to like-named capacitors shown in FIG. 2M above. Inone or more embodiments, one or more of the modules and elements shownin FIG. 4A may be omitted, repeated, and/or substituted. Accordingly,embodiments of the invention should not be considered limited to thespecific arrangements of modules shown in FIG. 4A.

FIG. 4B shows a schematic diagram of an example RF power sourceconnected to an equivalent circuit in accordance with one or moreembodiments of the invention. In particular, the example RF power source(108) shown in FIG. 4B may operate based on the ISM band as the powersource (108) depicted in FIGS. 1A, 1C, 2B, 2C, 2D, 2K, and 2L above.Specifically, the example RF power source (108) shown in FIG. 4Bincludes the terminal A (204 a) and terminal B (204 b) that correspondto the two terminals of the power source (108) depicted in FIGS. 1A, 1C,2B, 2C, 2D, 2K, and 2L above. The schematic diagram includes capacitors,inductors, and resistors of various RLC circuit components andcommercial part numbers of various integrated circuit components. Inparticular, the equivalent circuit (206 b) represents the rectangularloop (206) or rectangular loop (206 a) shown in FIGS. 2K and 2L above.In one or more embodiments, one or more of the modules and elementsshown in FIG. 4B may be omitted, repeated, and/or substituted.Accordingly, embodiments of the invention should not be consideredlimited to the specific arrangements of modules shown in FIG. 4B.

FIG. 5A shows a schematic diagram of an example receiver device A (500a) in accordance with one or more embodiments of the invention. In oneor more embodiments, one or more of the modules and elements shown inFIG. 5A may be omitted, repeated, and/or substituted. Accordingly,embodiments of the invention should not be considered limited to thespecific arrangements of modules shown in FIG. 5A.

As shown in FIG. 5A, the receiver device A (500 a) includes multiplelight emitting diodes (LEDs) (e.g., LED (502)) that are connected inparallel to form an LED string. The two ends of the LED string areconnected to a rectifier circuit A (501 a) to form a loop. For example,the loop may be a circular loop used as a mobile LED lighting deviceused within the wireless power transfer area (101) depicted in FIG. 1Aabove. In one or more embodiments of the invention, the rectifiercircuit A (501 a) includes capacitors C₁, C₂, and C₃ and rectifyingdiodes D₁ and D₂. When the receiver device A (500 a) is in the presenceof the oscillating magnetic fields, the changing magnetic flux throughthe loop of the LED string induces a voltage difference between the twoends of the LED string. The induced voltage difference oscillates withtime. The capacitance C₃ is adjusted to bring the LED string intoresonance with the oscillating magnetic fields to enhance the inducedoscillating voltage. The rectifying diodes D₁ and D₂ rectify the inducedoscillating voltage to produce a DC voltage difference between the outerwire (503 a) and inner wire (503 b) of the LED string thereby deliverpower to the parallel-connected LEDs (e.g., LED (502)). The capacitorsC₁ and C₂ act as RF bypass capacitors to maintain the outer wire (503 a)and inner wire (503 b) of the LED string appear shorted to the RFcurrent. The configuration of the receiver device A (500 a) limits theloop voltage by the combined forward voltage drop across the LEDs inseries with the rectifying diode D₁ or D₂, which improves safety to theuser.

Similar to FIG. 5A, FIG. 5B shows an example receiver device B (500 b),which is a larger version of the receiver device A (500 a) that hasmultiple rectifier circuits (i.e., rectifier circuit B (501 b),rectifier circuit C (501 c), rectifier circuit D (501 d), rectifiercircuit E (501 e)). The operation of the receiver device B (500 b) issubstantially the same as the receiver device A (500 a). The number ofsegments in the receiver device B (500 b) may be chosen to provide anoptimal impedance match to the load, i.e., the parallel-connected LEDs.

In addition to FIGS. 5A and 5B, FIG. 5C shows a schematic diagram ofother example receiver devices.

FIG. 5C shows a schematic diagram of an example receiver device circuit(500 c) in accordance with one or more embodiments of the invention. Inone or more embodiments, the receiver device circuit (500 c) is employedin various types of receiver devices having different shapes, sizes,form factors, etc. for various different types of mobile or stationeryapplications within the wireless power transfer area (101) depicted inFIG. 1A above. In one or more embodiments, at least the inductor, L, ofthe receiver device circuit (500 c) is placed within the wireless powertransfer area (101) for receiving the wireless power transfer. Theremaining components shown in FIG. 5C are configured to convert thereceived wireless power to suitable format to be consumed by a load,represented by the resistance, R_(L).

As shown in FIG. 5C, the inductor, L, along with capacitors, C₁, C₂, andC₃, are tuned to resonate at the characteristic frequency of thevariable form factor transmitter (102) and the RF power source (108)described in reference to FIGS. 1A through 2G above. The value ofcapacitor C₁ is chosen to provide an impedance match between theresonant receiver and the input of the DC-to-DC converter (504). TheDC-to-DC converter (504) transforms the rectified voltage into aconstant voltage to drive the load, R_(L). The DC-to-DC converter (504)allows the receiver device circuit (500 c) to present a constant voltageto the load R_(L) even in situations where the receiver device circuit(500 c) is moved through regions of varying magnetic field strengthwithin the wireless power transfer area (101). Note that the load R_(L)need not be a linear device, i.e., a device with a linear voltage versuscurrent relation. Examples of load R_(L) include, but are not limitedto, LED's, microcontrollers, motors, sensors, actuators, etc.

FIG. 5D shows a schematic diagram of an additional example receiverdevice circuit (500 d) in accordance with one or more embodiments of theinvention. The inductor, L, along with capacitors, C₁ and C₂, are tunedto resonate at the characteristic frequency of the variable form factortransmitter (102) and the RF power source (108) described in referenceto FIGS. 1A through 2G above. The value of capacitor C₁ is chosen toprovide an impedance match between the resonant receiver and the LEDload. The bridge rectifier converts the RF voltage present on capacitorC₁ into a DC voltage, which drives the LED. For example, the LED maycorrespond to the string of decorative light emitting diodes (LEDs)attached to the bottoms of glasses, depicted in FIG. 2N above.

FIG. 5E shows a layout diagram (500 e) of the example receiver devicecircuit (500 d) depicted in FIG. 5D above. The inductor, L, is composedof a conducting trace on the surface of a printed circuit board (PCB) inthe form of a flat spiral with multiple turns. Capacitors, C₁ and C₂,are placed in series with this spiral at the location (501). A secondlayer of traces is used on the PCB to allow connections to jump overmultiple turns of the inductor, L. Note also that C₁ and C₂ can beplaced in series with the turns of the inductor, L, at any point. InFIG. 5E, for example, the capacitor, C₂, is placed across a break in thecenter of the inductor, L. This placement helps to maintain symmetry inthe distribution of voltage on the turns of the inductor, L.

In one or more embodiments of the invention, the receiver device A (500a), receiver device B (500 b), receiver device circuit (500 c), orreceiver device circuit (500 d) may receive power wirelessly from anyelectromagnetic transmitter, such as a dipole transmitter (e.g.,magnetic dipole transmitter), a loop antenna with distributedcapacitance, a parallel-wire transmission line with distributedcapacitance, a shielded transmission line with distributed capacitance,etc. In one or more embodiments of the invention, the receiver device A(500 a), receiver device B (500 b), receiver device circuit (500 c),and/or or receiver device circuit (500 d) are placed within the wirelesspower transfer area (101) as the receiver device (A), receiver device(B), receiver device (C), receiver device (D), receiver device (E), orreceiver device (F) to receive power wirelessly from the variable formfactor transmitter (102).

FIGS. 6A-6F show schematic and layout diagrams, according to legend(600), of variations of the variable form factor transmitter (102)depicted in FIGS. 1A-1C above. In one or more embodiments of theinvention, the variable form factor transmitter (102) shown in FIGS.6A-6F may include, or otherwise be based on, example constructionsdescribed in reference to FIGS. 2A-2P above. For clarity, thedistributed capacitors may not be explicitly shown in FIGS. 6A-6F. Inone or more embodiments, the wireless power transfer area (101) shown inFIGS. 6A-6F may be a large wireless power transfer area. A largewireless power transfer area, or simply large area, has a dimension(e.g., length, width, diameter, etc.) that exceeds a wavelengthcorresponding to the aforementioned characteristic frequency of thevariable form factor transmitter (102).

In one or more embodiments of the invention, the variable form factortransmitter (102) shown in FIGS. 6A-6F may be arranged in a2-dimensional spatially-periodic structure to suppress far-fieldradiation due to oscillating current density. In particular, suppressingfar-field radiation in the variable form factor transmitter (102) shownin FIGS. 6A-6F is described in reference to FIGS. 7A-7F below.

In one or more embodiments, one or more of the modules and elementsshown in FIGS. 6A-6F may be omitted, repeated, and/or substituted.Accordingly, embodiments of the invention should not be consideredlimited to the specific arrangements of modules shown in FIGS. 6A-6F.

As shown in FIG. 6A, the variable form factor transmitter (102) includesa number of cross-coupled segments (e.g., segment A (601), segment B(601 b), segment C (601 c), etc.) disposed about the wireless powertransfer area (101). As used herein, a segment is a contiguous portionof the variable form factor transmitter (102). For example, the segmentmay be a section of the distributed capacitor string (207) depicted inFIG. 2F above. In one or more embodiments, the segment is arranged intoa loop form factor substantially enclosing a loop area and has at leastone pair of terminals for electrical connection to an adjacent segment(referred to as a neighbor). Adjacent segments are two segments withoutany intervening segment disposed between them. In one or moreembodiments, adjacent segments are electrically connected to each otherto pass electrical current from one segment to the other segment. Forexample, the segment A (601) has one pair of terminals (602 a) connectedto another pair of terminals (602 b) of the segment B (601 b). Inanother example, the segment B (601 b) has two pairs of terminals forconnecting to two adjacent segments (i.e., segment A (601 a) and segmentC (601 c)). In one or more embodiments, the electrical current flowingin each segment induces magnetic field substantially orthogonal to theloop area of the segment. The direction of the induced magnetic field isdependent on the rotational direction of the electrical current flowingthrough the segment. The rotational direction is one of clockwisedirection and counter clockwise direction. In one or more embodiments,the current is alternating current with directions of all currentsreversed during half of the cycle. The direction of the induced magneticfield is related to the rotational direction of the flowing currentaccording to a right-hand rule of the electromagnetic theorem.

In one or more embodiments, the segments of the variable form factortransmitter (102) are configured to transmit, from a radio frequency(RF) power source and based at least in part on the aforementionedcharacteristic frequency, RF power across the wireless power transferarea (101) via a near electromagnetic field of the variable form factortransmitter (102).

In one or more embodiments, each segment includes one or more sideswhere adjacent sides of the adjacent segments are configured to conductelectrical current in opposing rotational directions. For example, theelectrical currents flow in counter clockwise direction in the segment A(601 a) and segment C (601 c) such that the induced magnetic fields arein the magnetic field direction A, which corresponds to the directionflowing out of the loop area of the segment A (601 a) and segment C (601c) toward the viewer of FIG. 6A. In contrast, the electrical currentflows in clockwise direction in the segment B (601 b) such that theinduced magnetic fields are in the magnetic field direction B, whichcorresponds to the direction flowing into the loop area of the segment B(601 b) away from the viewer of FIG. 6A. As a result of the opposingdirections of magnetic fields induced by adjacent segments, a radiationloss of the wireless power transfer due to a far electromagnetic fieldof the variable form factor transmitter (102) is reduced. The repetitivepattern of opposing magnetic field directions (i.e., out-of-phasedirections) in adjacent segments is referred to as a checkerboard phasepattern. FIG. 6A shows a one dimensional checkerboard pattern. Examplesof two dimensional checkerboard patterns are shown in FIGS. 6E-6G below.

In one or more embodiments, the RF power source includes multiplephase-locked amplifiers disposed in at least a portion of the segmentsof the variable form factor transmitter (102). For example, each segmentshown in FIG. 6A includes a phase-locked RF amplifier. In one or moreembodiments, phase-locking among the segments is accomplished using aphase-locked loop and a master-slave topology, in which all phase-lockedRF amplifier share a master clock signal. While FIG. 6A depictedconnected segments forming a single loop, in an alternativeconfiguration having disconnected but coupled loops, each loop may havea phase-locked RF amplifier that measures its phase difference from itsneighbors, and adjusts its phase to be opposite to the mean phase of itsnearest neighbors. In this case, the multiple phase-locked amplifiers ofthe variable form factor transmitter (102) automatically arrangethemselves into a checkerboard pattern of alternating phases, withoutthe need for any centralized control. As used herein, alternating phasesrefers to opposing rotational directions of electrical currents flowingin adjacent segments.

In one or more embodiments, the segments of the variable form factortransmitter (102) have same shapes and/or dimensions. In one or moreembodiments, one or more segments of the variable form factortransmitter (102) may have different shapes and/or dimensions ascompared to remaining segments in the variable form factor transmitter(102). In one or more embodiments, the segments of the variable formfactor transmitter (102) are disposed in one dimensional ormulti-dimensional repetitive structure.

FIG. 6B shows a schematic layout diagram of a variation of the examplevariable form factor transmitter (102) depicted in FIG. 6A above wherethe RF power source includes a single RF amplifier. As shown in FIG. 6B,each segment is associated with multiple magnetic direction symbols toillustrate substantially uniform magnetic field across the loop area.

FIG. 6C shows a schematic layout diagram of a variation of the examplevariable form factor transmitter (102) depicted in FIG. 6A above wherethe segments are partitioned into two separate portions denoted asvariable form factor transmitter A (102 a) and variable form factortransmitter B (102 b) that are individually powered by separate RFamplifiers denoted as “A” and “B”. In one or more embodiments, thevariable form factor transmitter A (102 a) and variable form factortransmitter B (102 b) are physically shifted by half a segment width,and driven with a 90 degree relative phase shift. In particular, thisarrangement eliminates the nulls in the vertical component of themagnetic field which exist in the arrangement shown in FIG. 6B. The 90degree phase shift does not have any effect on the radiation loss.

FIG. 6D shows a schematic layout diagram of a variation of the examplevariable form factor transmitter (102) depicted in FIG. 6C above wherethe variable form factor transmitter A (102 a) is not explicitly drivenby an RF amplifier. Instead, the induced magnetic fields from thevariable form factor transmitter B (102 b) in turn induces electriccurrent to flow in the segments of the variable form factor transmitterA (102 a). Although the variable form factor transmitter A (102 a) andvariable form factor transmitter B (102 b) are independent of each otherin terms of physical and electrical connections, they becomemagnetically coupled resonators.

In the case where the variable form factor transmitter (102) isconstructed from multiple magnetically coupled segments, the resonantproperties of the combined structure may be used to ensure the correctphase relation exists between the magnetically coupled segments withoutthe need for active phase control. The effect employed in this case isthe frequency splitting which develops between any two or more coupledresonators. When the resonators are arranged in a periodic structure,the resonant eigen-frequencies approach the form of a continuous band asthe size of the structure is extended without bound. Bloch's theoremapplies to such a structure, and the eigen-modes of excitation aredescribed by Bloch wavefunctions. The eigen-mode with the highestspatial frequency has the lowest temporal frequency. This highestspatial-frequency eigen-mode corresponds to the desired checkerboardphase pattern. It is therefore possible to ensure the correct phaserelation between individual variable form factor transmitters by drivingthe structure at its lowest temporal eigen-frequency within the Blochband.

In one or more embodiments, the wireless power transfer operates at afixed, pre-determined frequency. The system of magnetically coupledsegments therefore are designed such that the lowest temporal-frequencyeigen-mode in the Bloch band has a resonant frequency equal to thedesired, pre-determined frequency of the wireless power transfer.

The system of magnetically coupled segments may be driven by anamplifier connected to a single segment, in which case the RF powerspreads through the structure to establish the checkerboard phasepattern described above, so long as the resonant conditions describedabove are met. However, if the magnetically coupled segments have anyradiation loss, the amplitude of the magnetic field decays geometricallywith every segment-to-segment hop. Depending on the size of the system,it may therefore be necessary to drive the system from multiple points.If this is the case, then each separate amplifier must be properlyphase-locked, so as to maintain the desired checkerboard phase pattern.

Note that when the total length of conductive wire in a single segmentapproaches a half of the free-space wavelength of the frequency ofoperation, there will be substantial charge build-up on the wirestructure due to its self-capacitance. The solution is to adddistributed capacitors at regular intervals in series with the segment.In other words, each segment is constructed as a distributed capacitorstring described above.

FIG. 6E shows a schematic layout diagram of a variation of the examplevariable form factor transmitter (102) depicted in FIG. 6A above wherethe segments have substantially rectangular shapes that are arranged tocover a large two dimensional geographic area. Note that the transmitterdepicted in FIG. 6E consists of multiple, disconnected loops of wirewhich are inductively coupled. Like the transmitter shown in FIG. 6D,this inductive coupling ensures that power is spread evenly throughoutthe structure.

FIG. 6F shows a schematic layout diagram of a variation of the examplevariable form factor transmitter (102) depicted in FIG. 6E above wherethe segments are connected according to an endless knot pattern.

FIG. 6G shows a schematic layout diagram of a variation of the examplevariable form factor transmitter (102) depicted in FIG. 6D above whereeach segment has three neighbors (i.e., adjacent segments), in contrastto the four-neighbor configurations depicted in FIGS. 6E and 6F above.

FIGS. 7A-7F illustrates suppression of far-field radiation by anoscillating current density arranged in a 2-dimensionalspatially-periodic structure. FIG. 7A shows a two-dimensional,spatially-periodic current distribution, lying entirely in the x-yplane. Let the vectors a₁ and a₂ denote the two primitive translationvectors of the structure.

The position-space current density function, J(x), has the property:

J(x+n ₁ a ₁ +n ₂ a ₂)=J(x)  (1)

where n₁ and n₂ are any integers. Equation 1 expresses the discretetranslational symmetry of the spatially-periodic structure.

The reciprocal lattice vectors, b₁ and b₂, may be written as functionsof the primitive translation vectors:

b ₁≡2πa ₂·(a ₁ ∧a ₂)⁻¹  (2)

b ₂≡−2πa ₁·(a ₁ ∧a ₂)⁻¹  (3)

where {circumflex over ( )} denotes the wedge product of two vectors.The reciprocal lattice vectors have the property:

b _(n) ·a _(m)=2πδ_(nm)  (4)

where δ_(nm) is the Kronecker delta function.

Additionally, let it be stipulated that the current density has theadditional symmetry:

$\begin{matrix}{{J\left( {{x \pm \frac{a_{1}}{2}} \pm \frac{a_{2}}{2}} \right)} = {- {J(x)}}} & (5)\end{matrix}$

The condition stipulated in Equation 5 ensures that the current densityexhibits a checkerboard-pattern translational symmetry, in addition tothe periodic translational symmetry described by Equation 1. Inspectionof FIG. 7A will show that the current distribution depicted thereinobeys the checkerboard-pattern translational symmetry described byEquation 5.

Let {tilde over (J)}(k) denote the spatial Fourier Transform of thecurrent density, where k is the spatial wavevector.

FIG. 7B shows the reciprocal lattice in Fourier space. The function,{tilde over (J)}(k), will be zero everywhere, except on lines ofsingularity extended infinitely, parallel to the k_(z) axis, andcentered on each of the lattice points of the reciprocal lattice in thek_(x)-k_(y) plane, as depicted in FIG. 7B.

The condition stipulated in Equation 5 implies the following conditionin Fourier space:

$\begin{matrix}{{{\overset{\sim}{J}(k)}e^{{- {ik}} \cdot {({{\pm \frac{a_{1}}{2}} \pm \frac{a_{2}}{2}})}}} = {- {\overset{\sim}{J}(k)}}} & (6) \\{{{\overset{\sim}{J}(k)}} = {\frac{1}{2}\left( {1 - e^{{- {ik}} \cdot {({{\pm \frac{a_{1}}{2}} \pm \frac{a_{2}}{2}})}}} \right){\overset{\sim}{J}(k)}}} & (7)\end{matrix}$

Equation 7 implies that {tilde over (J)}(k) must be zero at any pointwhere the dot-product, k·(±a₁/2±a₂/2), is an integer multiple of 2π. Atthe lattice points of the reciprocal lattice, the wavevector, k, may beexpressed as:

k=n ₁ b ₁ +n ₂ b ₂  (8)

where n₁ and n₂ are integers. Therefore, {tilde over (J)}(k) will bezero at every lattice point of the reciprocal lattice where thefollowing condition holds:

$\begin{matrix}{{{\pm \frac{n_{1}}{2}} \pm \frac{n_{2}}{2}} \in {\mathbb{Z}}} & (9)\end{matrix}$

Condition 9 will be satisfied at all lattice points where the integersn₁ and n₂ are either both odd or both even. Therefore, the only latticepoints in the reciprocal lattice on which {tilde over (J)}(k) can take anon-zero value are those where one of the two integers is odd, and theother is even.

FIG. 7C shows the points in the k_(x)-k_(y) plane at which {tilde over(J)}(k) may take a non-zero value. The function {tilde over (J)}(k)consists of lines of singularity, centered at these points in thek_(x)-k_(y) plane and extended infinitely along the k_(z) axis.

Assume that the current distribution, J(x), oscillates sinusoidally intime with angular frequency, ω. Let c represent the speed of light inthe surrounding medium, and let κ=ω/c represent the free-spacewavenumber of a propagating electromagnetic wave with angular frequency,ω. The power radiated by the current density, J(x), is given by thefollowing integral performed in Fourier space:

$\begin{matrix}{P_{rad} = {\frac{{\zeta\kappa}^{2}}{32\; \pi^{2}}{\int{d\; \Omega \overset{\sim}{J}*{(k) \cdot {\overset{\sim}{J}(k)}}}}}} & (10)\end{matrix}$

where k=κ{circumflex over (r)}, where {circumflex over (r)} is thedirection unit vector of the outgoing radiation, and the angularintegral, ∫dΩ, is taken over all directions of {circumflex over (r)}.Note that Equation 10 is only valid for distributions of current whichare divergence-free, i.e. ∇·J=0.

The integral in Equation 10 is performed over the surface of a sphere inFourier space, of radius κ, centered on the origin. FIGS. 7D and 7E showthe circular cross-section of this surface in the k_(x)-k_(y) plane forcase (a) and case (b), respectively. In particular, FIG. 7D shows anexample case (a) in which far-field radiation is suppressed. In case(a), the magnitude of the wavenumber, κ, is less than both |b₁| and|b₂|. In contrast, FIG. 7E shows an example case (b) in which far-fieldradiation is not suppressed. In case (b), the magnitude of κ is largerthan one of |b₁| and |b₂|.

In case (a), the sphere of radius κ does not intersect any of the linesof singularity on which {tilde over (J)}(k) takes on a non-zero value.Therefore, the value of the function {tilde over (J)}(k) is zeroeverywhere on the surface of this sphere, and the integral of Equation10 is exactly zero for case (a).

In case (b), some of the lines of singularity, where {tilde over (J)}(k)takes on a non-zero value, intersect the surface of the sphere of radiusκ. Therefore, the integral of Equation 10 is non-zero for case (b).

Accordingly, the radiated power will be suppressed as long as thefollowing condition is met:

κ<min(|b ₁ |,|b ₂|)  (11)

Note that the checkerboard-pattern translational symmetry, expressed byEquation 5, ensures the suppression of far-field radiation which occurswhen Equation 11 is satisfied. This is because the checkerboard-patterntranslational symmetry forces {tilde over (J)}(k) to have zero amplitudeat all of the lattice points defined by Equation 9, including theorigin. If the amplitude of {tilde over (J)}(k) were not zero at theorigin of the reciprocal lattice, then {tilde over (J)}(k) would have aline of singularity extending along the k_(z) axis, which wouldintersect a sphere centered on the origin, regardless of the radius ofthe sphere. Therefore, the integral in Equation 10 would be non-zero forall values of κ.

FIG. 7F shows an example of a rectangular checkerboard grid where thecurrent density is arranged. The two primitive translation vectors are:

a ₁ =we ₁ +he ₂  (12)

a ₂ =−we ₁ +he ₂  (13)

where w and h are the width and height of the rectangular cells, andwhere e₁ and e₂ are unit vector pointing in the x and y directions,respectively. The reciprocal lattice vectors are:

$\begin{matrix}{b_{1} = {{\frac{\pi}{\omega}e_{1}} + {\frac{\pi}{h}e_{2}}}} & (14) \\{b_{2} = {{{- \frac{\pi}{\omega}}e_{1}} + {\frac{\pi}{h}e_{2}}}} & (15)\end{matrix}$

Both reciprocal lattice vectors have the same magnitude, given by:

$\begin{matrix}{{b_{1}} = {{b_{2}} = {\pi \sqrt{\frac{1}{\omega^{2}} + \frac{1}{h^{2}}}}}} & (16)\end{matrix}$

The condition for the suppression of far-field radiation can beexpressed as:

$\begin{matrix}{{\kappa < {\min \left( {{b_{1}},{b_{2}}} \right)}} = {\pi \sqrt{\frac{1}{\omega^{2}} + \frac{1}{h^{2}}}}} & (17)\end{matrix}$

This condition may also be expressed in terms of the free-spacewavelength, λ=2π/κ:

$\begin{matrix}{\lambda > \frac{2}{\sqrt{\frac{1}{\omega^{2}} + \frac{1}{h^{2}}}}} & (18)\end{matrix}$

In the case where h>>w, the rectangular grid approaches a zebra-stripepattern, and the condition for radiation suppression becomes:

λ>2w  (19)

FIG. 8 shows a method flowchart in accordance with one or moreembodiments of the invention. In one or more embodiments, the method maybe based on one or more variable form factor transmitters depicted inFIGS. 6A-6G above. One or more steps shown in FIG. 8 may be omitted,repeated, and/or performed in a different order among differentembodiments of the invention. Accordingly, embodiments of the inventionshould not be considered limited to the specific number and arrangementof steps shown in FIG. 8.

Initially in Step 801, a variable form factor transmitter is adaptedinto at least a number of cross-coupled segments disposed about apre-determined wireless power transfer area. The pre-determined wirelesspower transfer area includes a dimension exceeding a wavelengthcorresponding to a characteristic frequency of the variable form factortransmitter. In one or more embodiments, one or more cross-coupledsegments is constructed using a string of distributed capacitors.

In Step 802, a number of phase-locked amplifiers are disposed in atleast a portion of the cross-coupled segments as a radio frequency (RF)power source for wireless power transfer.

In Step 803, from the RF power source and based at least in part on acharacteristic frequency, RF power is transmitted across thepre-determined wireless power transfer area via a near electromagneticfield of the variable form factor transmitter.

In Step 804, based on opposing directions of magnetic fields induced byadjacent cross-coupled segments, a radiation loss of the wireless powertransfer due to a far electromagnetic field of the variable form factortransmitter is reduced.

In Step 805, receiver devices are disposed within the pre-determinedwireless power transfer area to receive the RF power transfer. Inparticular, a portion of the RF power transmitted via the variable formfactor transmitter from the RF power source is received by the receiverdevices, where the characteristic frequency is substantially independentof a number or placement of the receiver devices.

While the invention has been described with respect to a limited numberof embodiments, those skilled in the art, having benefit of thisdisclosure, will appreciate that other embodiments can be devised whichdo not depart from the scope of the invention as disclosed herein.Accordingly, the scope of the invention should be limited only by theattached claims.

What is claimed is:
 1. A method for wireless power transfer, comprising:adapting a variable form factor transmitter into at least a plurality ofcross-coupled segments disposed about a pre-determined wireless powertransfer area; transmitting, from a radio frequency (RF) power sourceand based at least in part on the characteristic frequency, RF poweracross the pre-determined wireless power transfer area via a nearelectromagnetic field of the variable form factor transmitter; andwherein, based on opposing directions of magnetic fields induced byadjacent cross-coupled segments of the plurality of cross-coupledsegments, a radiation loss of the wireless power transfer is reduced dueto a far electromagnetic field of the variable form factor transmitter.2. The method of claim 1, wherein each of the plurality of cross-coupledsegments comprises a plurality of sides; and wherein adjacent sides ofthe adjacent cross-coupled segments are configured to conduct electricalcurrent in opposing rotational directions.
 3. The method of claim 1,further comprising: disposing a plurality of phase-locked amplifiers inat least a portion of the plurality of cross-coupled segments as the RFpower source.
 4. The method of claim 1, wherein the plurality ofcross-coupled segments are disposed about the pre-determined wirelesspower transfer area according to a pattern of endless knot.
 5. Themethod of claim 1, wherein the pre-determined wireless power transferarea comprises a dimension exceeding a wavelength corresponding to acharacteristic frequency of the variable form factor transmitter.
 6. Themethod of claim 1, further comprising: forming one or more of theplurality of cross-coupled segments using a string of distributedcapacitors; connecting a plurality of capacitors in series via at leasta plurality of wire segments into the string of distributed capacitors;wherein each of the plurality of capacitors comprises a pre-determinedcapacitance; wherein each of the plurality of wire segments comprises apre-determined segment length and a pre-determined inductance per unitlength; and wherein the characteristic frequency is dependent on atleast the pre-determined capacitance and the pre-determined inductanceper unit length.
 7. The method of claim 1, further comprising: disposinga plurality of receiver devices within the pre-determined wireless powertransfer area; wherein a portion of the RF power transmitted via thevariable form factor transmitter from the RF power source is received bythe plurality of receiver devices, and wherein the characteristicfrequency is substantially independent of a number or placement of theplurality of receiver devices.
 8. A transmitter for wireless powertransfer, comprising: a plurality of cross-coupled segments disposedabout a pre-determined wireless power transfer area; wherein theplurality of cross-coupled segments are configured to: transmit, from aradio frequency (RF) power source and based at least in part on thecharacteristic frequency, RF power across the pre-determined wirelesspower transfer area via a near electromagnetic field of the transmitter;and reduce, based on opposing directions of magnetic fields induced byadjacent cross-coupled segments of the plurality of cross-coupledsegments, a radiation loss of the wireless power transfer due to a farelectromagnetic field of the transmitter.
 9. The transmitter of claim 8,wherein each of the plurality of cross-coupled segments comprises aplurality of sides; and wherein adjacent sides of the adjacentcross-coupled segments are configured to conduct electrical current inopposing rotational directions.
 10. The transmitter of claim 8, the RFpower source comprising: a plurality of phase-locked amplifiers disposedin at least a portion of the plurality of cross-coupled segments. 11.The transmitter of claim 8, wherein the plurality of cross-coupledsegments are disposed about the pre-determined wireless power transferarea according to a pattern of endless knot.
 12. The transmitter ofclaim 8, wherein the transmitter is a variable form factor transmitter;and wherein the pre-determined wireless power transfer area comprises adimension exceeding a wavelength corresponding to a characteristicfrequency of the variable form factor transmitter.
 13. The transmitterof claim 8, one or more of the plurality of cross-coupled segmentscomprising: a plurality of capacitors each with a pre-determinedcapacitance; a plurality of wire segments each with a pre-determinedsegment length and a pre-determined inductance per unit length; whereinthe plurality of capacitors are connected in series via at least theplurality of wire segments into a string of distributed capacitors; andwherein the characteristic frequency is dependent on at least thepre-determined capacitance and the pre-determined inductance per unitlength.
 14. A system for wireless power transfer, comprising: aplurality of cross-coupled segments disposed about a pre-determinedwireless power transfer area; and a radio frequency (RF) power sourcecoupled to the plurality of cross-coupled segments, wherein theplurality of cross-coupled segments are configured to: transmit, fromthe RF power source and based at least in part on the characteristicfrequency, RF power across the pre-determined wireless power transferarea via a near electromagnetic field of the plurality of cross-coupledsegments; and reduce, based on opposing directions of magnetic fieldsinduced by adjacent cross-coupled segments of the plurality ofcross-coupled segments, a radiation loss of the wireless power transferdue to a far electromagnetic field of the plurality of cross-coupledsegments.
 15. The system of claim 14, wherein each of the plurality ofcross-coupled segments comprises a plurality of sides; and whereinadjacent sides of the adjacent cross-coupled segments are configured toconduct electrical current in opposing rotational directions.
 16. Thesystem of claim 14, the RF power source comprising: a plurality ofphase-locked amplifiers disposed in at least a portion of the pluralityof cross-coupled segments.
 17. The system of claim 14, wherein theplurality of cross-coupled segments are disposed about thepre-determined wireless power transfer area according to a pattern ofendless knot.
 18. The system of claim 14, wherein the pre-determinedwireless power transfer area comprises a dimension exceeding awavelength corresponding to a characteristic frequency of the pluralityof cross-coupled segments.
 19. The system of claim 14, one or more ofthe plurality of cross-coupled segments comprising: a plurality ofcapacitors each with a pre-determined capacitance; a plurality of wiresegments each with a pre-determined segment length and a pre-determinedinductance per unit length; wherein the plurality of capacitors areconnected in series via at least the plurality of wire segments into astring of distributed capacitors; and wherein the characteristicfrequency is dependent on at least the pre-determined capacitance andthe pre-determined inductance per unit length.
 20. The system of claim14, further comprising: a plurality of receiver devices disposed withinthe pre-determined wireless power transfer area; wherein a portion ofthe RF power transmitted via the plurality of cross-coupled segmentsfrom the RF power source is received by the plurality of receiverdevices; and wherein the characteristic frequency is substantiallyindependent of a number or placement of the plurality of receiverdevices.